How to Win Coding Competitions: Secrets of Champions

Week 4: Algorithms on GraphsLecture 9: Single Source Shortest Paths

Maxim BuzdalovSaint Petersburg 2016

Single Source Shortest Path problem

Problem: for every vertex determine a shortest path from v0

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Single Source Shortest Path problem

Problem: for every vertex determine a shortest path from v0

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Single Source Shortest Path problem

Problem: for every vertex determine a shortest path from v0

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example.

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)

Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example.

0

∞

∞

∞

∞ ∞

∞

∞ ∞

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∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example.

0

∞

∞

∞

∞ ∞

∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example.

0

∞

∞

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∞ ∞

∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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∞

3

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∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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∞ ∞

∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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∞ ∞

∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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∞ ∞

∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

0

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∞ ∞

∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

0

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∞ ∞

∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

0

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∞ ∞

∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

0

4

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∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 1

0

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∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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∞

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 2

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3

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Bellman-Ford algorithm

Bellman-Ford algorithm:I Do |V | − 1 times:

I Visit all edgesI For (u, v), update

shortest distance to vI If nothing updated,

terminate

I Running time: O(|V | · |E |)Why does it work?

I First k iterations buildfirst k segments in everyshortest path

Example. Iteration 3. Nothing changed. We may stop

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Bellman-Ford algorithm and negative cycles

A negative cycle is a cycle of total negative length

I If any vertex of the negative cycle is reachable, then there is no shortest distanceto any vertex of this cycle, and any vertex reachable from it

The Bellman-Ford algorithm can detect negative cycles. How?

I Update shortest distances along all edges once more

I If a shortest distance to v changes, then v is reachable from a negative cycle

4 / 8

Bellman-Ford algorithm and negative cycles

A negative cycle is a cycle of total negative length

I If any vertex of the negative cycle is reachable, then there is no shortest distanceto any vertex of this cycle, and any vertex reachable from it

The Bellman-Ford algorithm can detect negative cycles. How?

I Update shortest distances along all edges once more

I If a shortest distance to v changes, then v is reachable from a negative cycle

4 / 8

Dijkstra algorithm

An efficient algorithm for non-negative edge lengths

I Idea: Maintain a set of vertices S with determined shortest distance from v0I Initially, S = {v0}I For all v /∈ S , maintain shortest distance estimation: D ′[v ] = minu∈S D[u] + L(u, v)

I Lemma: For a v /∈ S with the smallest D ′[v ], the shortest distance is D ′[v ]

I Assume it is not trueI There is another path which yields a distance strictly smaller than D ′[v ]I It should go through other v ′ /∈ SI But edge lengths are non-negative → contradiction

I How to update S :I Choose v /∈ S with the smallest D ′[v ]I Set D[v ] = D ′[v ]I S ← S ∪ {v}I For all edges (v , v ′) ∈ E , update D ′[v ′]← min(D ′[v ′],D[v ] + L(v , v ′))

5 / 8

Dijkstra algorithm

An efficient algorithm for non-negative edge lengths

I Idea: Maintain a set of vertices S with determined shortest distance from v0I Initially, S = {v0}I For all v /∈ S , maintain shortest distance estimation: D ′[v ] = minu∈S D[u] + L(u, v)

I Lemma: For a v /∈ S with the smallest D ′[v ], the shortest distance is D ′[v ]

I Assume it is not trueI There is another path which yields a distance strictly smaller than D ′[v ]I It should go through other v ′ /∈ SI But edge lengths are non-negative → contradiction

I How to update S :I Choose v /∈ S with the smallest D ′[v ]I Set D[v ] = D ′[v ]I S ← S ∪ {v}I For all edges (v , v ′) ∈ E , update D ′[v ′]← min(D ′[v ′],D[v ] + L(v , v ′))

5 / 8

Dijkstra algorithm

An efficient algorithm for non-negative edge lengths

I Idea: Maintain a set of vertices S with determined shortest distance from v0I Initially, S = {v0}I For all v /∈ S , maintain shortest distance estimation: D ′[v ] = minu∈S D[u] + L(u, v)

I Lemma: For a v /∈ S with the smallest D ′[v ], the shortest distance is D ′[v ]

I Assume it is not trueI There is another path which yields a distance strictly smaller than D ′[v ]I It should go through other v ′ /∈ SI But edge lengths are non-negative → contradiction

I How to update S :I Choose v /∈ S with the smallest D ′[v ]I Set D[v ] = D ′[v ]I S ← S ∪ {v}I For all edges (v , v ′) ∈ E , update D ′[v ′]← min(D ′[v ′],D[v ] + L(v , v ′))

5 / 8

Dijkstra algorithm

An efficient algorithm for non-negative edge lengths

I Idea: Maintain a set of vertices S with determined shortest distance from v0I Initially, S = {v0}I For all v /∈ S , maintain shortest distance estimation: D ′[v ] = minu∈S D[u] + L(u, v)

I Lemma: For a v /∈ S with the smallest D ′[v ], the shortest distance is D ′[v ]I Assume it is not trueI There is another path which yields a distance strictly smaller than D ′[v ]

I It should go through other v ′ /∈ SI But edge lengths are non-negative → contradiction

I How to update S :I Choose v /∈ S with the smallest D ′[v ]I Set D[v ] = D ′[v ]I S ← S ∪ {v}I For all edges (v , v ′) ∈ E , update D ′[v ′]← min(D ′[v ′],D[v ] + L(v , v ′))

5 / 8

Dijkstra algorithm

An efficient algorithm for non-negative edge lengths

I Idea: Maintain a set of vertices S with determined shortest distance from v0I Initially, S = {v0}I For all v /∈ S , maintain shortest distance estimation: D ′[v ] = minu∈S D[u] + L(u, v)

I Lemma: For a v /∈ S with the smallest D ′[v ], the shortest distance is D ′[v ]I Assume it is not trueI There is another path which yields a distance strictly smaller than D ′[v ]I It should go through other v ′ /∈ S

I But edge lengths are non-negative → contradictionI How to update S :

I Choose v /∈ S with the smallest D ′[v ]I Set D[v ] = D ′[v ]I S ← S ∪ {v}I For all edges (v , v ′) ∈ E , update D ′[v ′]← min(D ′[v ′],D[v ] + L(v , v ′))

5 / 8

Dijkstra algorithm

An efficient algorithm for non-negative edge lengths

I Idea: Maintain a set of vertices S with determined shortest distance from v0I Initially, S = {v0}I For all v /∈ S , maintain shortest distance estimation: D ′[v ] = minu∈S D[u] + L(u, v)

I Lemma: For a v /∈ S with the smallest D ′[v ], the shortest distance is D ′[v ]I Assume it is not trueI There is another path which yields a distance strictly smaller than D ′[v ]I It should go through other v ′ /∈ SI But edge lengths are non-negative → contradiction

I How to update S :I Choose v /∈ S with the smallest D ′[v ]I Set D[v ] = D ′[v ]I S ← S ∪ {v}I For all edges (v , v ′) ∈ E , update D ′[v ′]← min(D ′[v ′],D[v ] + L(v , v ′))

5 / 8

Dijkstra algorithm

An efficient algorithm for non-negative edge lengths

I Idea: Maintain a set of vertices S with determined shortest distance from v0I Initially, S = {v0}I For all v /∈ S , maintain shortest distance estimation: D ′[v ] = minu∈S D[u] + L(u, v)

I Lemma: For a v /∈ S with the smallest D ′[v ], the shortest distance is D ′[v ]I Assume it is not trueI There is another path which yields a distance strictly smaller than D ′[v ]I It should go through other v ′ /∈ SI But edge lengths are non-negative → contradiction

I How to update S :I Choose v /∈ S with the smallest D ′[v ]I Set D[v ] = D ′[v ]I S ← S ∪ {v}I For all edges (v , v ′) ∈ E , update D ′[v ′]← min(D ′[v ′],D[v ] + L(v , v ′))

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Example run

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Dijkstra algorithm. Implementation choices

I Recall: How to update SI Choose v /∈ S with the smallest D ′[v ]I Set D[v ] = D ′[v ]I S ← S ∪ {v}I For all edges (v , v ′) ∈ E , update D ′[v ′]← min(D ′[v ′],D[v ] + L(v , v ′))

I How to choose vertices with smallest distance?1. Näıve way. Iterate each time over vertices

I Running time of one iteration: O(|V |)I |V | iterations, O(|V |2) total time. Good for dense graphs, bad for sparse ones

2. Using binary heapI Choosing – “extract minimum”: O(log |V |), at most |V | operationsI Updating – “decrease key”: O(log |V |), at most |E | operationsI Total running time: O(|E | log |V |). Good for sparse graphs, bad for dense ones

3. Using Fibonacci heap: “decrease key” in amortized O(1) timeI Total running time: O(|V | log |V | + |E |). However, impractical :(

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Dijkstra algorithm. Implementation choices

I Recall: How to update SI Choose v /∈ S with the smallest D ′[v ]I Set D[v ] = D ′[v ]I S ← S ∪ {v}I For all edges (v , v ′) ∈ E , update D ′[v ′]← min(D ′[v ′],D[v ] + L(v , v ′))

I How to choose vertices with smallest distance?

1. Näıve way. Iterate each time over verticesI Running time of one iteration: O(|V |)I |V | iterations, O(|V |2) total time. Good for dense graphs, bad for sparse ones

2. Using binary heapI Choosing – “extract minimum”: O(log |V |), at most |V | operationsI Updating – “decrease key”: O(log |V |), at most |E | operationsI Total running time: O(|E | log |V |). Good for sparse graphs, bad for dense ones

3. Using Fibonacci heap: “decrease key” in amortized O(1) timeI Total running time: O(|V | log |V | + |E |). However, impractical :(

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Dijkstra algorithm. Implementation choices

I Recall: How to update SI Choose v /∈ S with the smallest D ′[v ]I Set D[v ] = D ′[v ]I S ← S ∪ {v}I For all edges (v , v ′) ∈ E , update D ′[v ′]← min(D ′[v ′],D[v ] + L(v , v ′))

I How to choose vertices with smallest distance?1. Näıve way. Iterate each time over vertices

I Running time of one iteration: O(|V |)I |V | iterations, O(|V |2) total time. Good for dense graphs, bad for sparse ones

2. Using binary heapI Choosing – “extract minimum”: O(log |V |), at most |V | operationsI Updating – “decrease key”: O(log |V |), at most |E | operationsI Total running time: O(|E | log |V |). Good for sparse graphs, bad for dense ones

3. Using Fibonacci heap: “decrease key” in amortized O(1) timeI Total running time: O(|V | log |V | + |E |). However, impractical :(

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Dijkstra algorithm. Implementation choices

I Recall: How to update SI Choose v /∈ S with the smallest D ′[v ]I Set D[v ] = D ′[v ]I S ← S ∪ {v}I For all edges (v , v ′) ∈ E , update D ′[v ′]← min(D ′[v ′],D[v ] + L(v , v ′))

I How to choose vertices with smallest distance?1. Näıve way. Iterate each time over vertices

I Running time of one iteration: O(|V |)I |V | iterations, O(|V |2) total time. Good for dense graphs, bad for sparse ones

2. Using binary heapI Choosing – “extract minimum”: O(log |V |), at most |V | operationsI Updating – “decrease key”: O(log |V |), at most |E | operationsI Total running time: O(|E | log |V |). Good for sparse graphs, bad for dense ones

3. Using Fibonacci heap: “decrease key” in amortized O(1) timeI Total running time: O(|V | log |V | + |E |). However, impractical :(

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Dijkstra algorithm. Implementation choices

I Recall: How to update SI Choose v /∈ S with the smallest D ′[v ]I Set D[v ] = D ′[v ]I S ← S ∪ {v}I For all edges (v , v ′) ∈ E , update D ′[v ′]← min(D ′[v ′],D[v ] + L(v , v ′))

I How to choose vertices with smallest distance?1. Näıve way. Iterate each time over vertices

I Running time of one iteration: O(|V |)I |V | iterations, O(|V |2) total time. Good for dense graphs, bad for sparse ones

2. Using binary heapI Choosing – “extract minimum”: O(log |V |), at most |V | operationsI Updating – “decrease key”: O(log |V |), at most |E | operationsI Total running time: O(|E | log |V |). Good for sparse graphs, bad for dense ones

3. Using Fibonacci heap: “decrease key” in amortized O(1) timeI Total running time: O(|V | log |V | + |E |). However, impractical :(

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Dijkstra with heap vs. Standard libraries

Contest trick: How to implement Dijkstra with heap using standard libraries?

I Libraries of C++ and Java do not support heaps with “decrease key” operation

I Basically two choices:1. Simulate binary heap with a binary tree of vertices ordered by distance

I “Extract minimum”: remove the minimum (leftmost) vertexI “Decrease key”: remove vertex, change distance, insert vertexI O(|E | log |V |) but quite slow due to tree’s hidden constant

2. Simulate binary heap with a priority queue of vertex-distance pairsI “Extract minimum”: remove the minimum record,

discard and continue if distance in pair differs from current distance to that vertexI “Decrease key”: just insert a vertex-distance pair with the new distanceI O(|E | log |E |) but rather fast, generally better than tree

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Dijkstra with heap vs. Standard libraries

Contest trick: How to implement Dijkstra with heap using standard libraries?

I Libraries of C++ and Java do not support heaps with “decrease key” operationI Basically two choices:

1. Simulate binary heap with a binary tree of vertices ordered by distanceI “Extract minimum”: remove the minimum (leftmost) vertexI “Decrease key”: remove vertex, change distance, insert vertexI O(|E | log |V |) but quite slow due to tree’s hidden constant

2. Simulate binary heap with a priority queue of vertex-distance pairsI “Extract minimum”: remove the minimum record,

discard and continue if distance in pair differs from current distance to that vertexI “Decrease key”: just insert a vertex-distance pair with the new distanceI O(|E | log |E |) but rather fast, generally better than tree

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Dijkstra with heap vs. Standard libraries

Contest trick: How to implement Dijkstra with heap using standard libraries?

I Libraries of C++ and Java do not support heaps with “decrease key” operationI Basically two choices:

1. Simulate binary heap with a binary tree of vertices ordered by distanceI “Extract minimum”: remove the minimum (leftmost) vertexI “Decrease key”: remove vertex, change distance, insert vertex

I O(|E | log |V |) but quite slow due to tree’s hidden constant2. Simulate binary heap with a priority queue of vertex-distance pairs

I “Extract minimum”: remove the minimum record,discard and continue if distance in pair differs from current distance to that vertex

I “Decrease key”: just insert a vertex-distance pair with the new distanceI O(|E | log |E |) but rather fast, generally better than tree

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Dijkstra with heap vs. Standard libraries

Contest trick: How to implement Dijkstra with heap using standard libraries?

I Libraries of C++ and Java do not support heaps with “decrease key” operationI Basically two choices:

1. Simulate binary heap with a binary tree of vertices ordered by distanceI “Extract minimum”: remove the minimum (leftmost) vertexI “Decrease key”: remove vertex, change distance, insert vertexI O(|E | log |V |) but quite slow due to tree’s hidden constant

2. Simulate binary heap with a priority queue of vertex-distance pairsI “Extract minimum”: remove the minimum record,

discard and continue if distance in pair differs from current distance to that vertexI “Decrease key”: just insert a vertex-distance pair with the new distanceI O(|E | log |E |) but rather fast, generally better than tree

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Dijkstra with heap vs. Standard libraries

Contest trick: How to implement Dijkstra with heap using standard libraries?

I Libraries of C++ and Java do not support heaps with “decrease key” operationI Basically two choices:

1. Simulate binary heap with a binary tree of vertices ordered by distanceI “Extract minimum”: remove the minimum (leftmost) vertexI “Decrease key”: remove vertex, change distance, insert vertexI O(|E | log |V |) but quite slow due to tree’s hidden constant

2. Simulate binary heap with a priority queue of vertex-distance pairsI “Extract minimum”: remove the minimum record,

discard and continue if distance in pair differs from current distance to that vertexI “Decrease key”: just insert a vertex-distance pair with the new distance

I O(|E | log |E |) but rather fast, generally better than tree

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Dijkstra with heap vs. Standard libraries

Contest trick: How to implement Dijkstra with heap using standard libraries?

I Libraries of C++ and Java do not support heaps with “decrease key” operationI Basically two choices:

1. Simulate binary heap with a binary tree of vertices ordered by distanceI “Extract minimum”: remove the minimum (leftmost) vertexI “Decrease key”: remove vertex, change distance, insert vertexI O(|E | log |V |) but quite slow due to tree’s hidden constant

2. Simulate binary heap with a priority queue of vertex-distance pairsI “Extract minimum”: remove the minimum record,

discard and continue if distance in pair differs from current distance to that vertexI “Decrease key”: just insert a vertex-distance pair with the new distanceI O(|E | log |E |) but rather fast, generally better than tree

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