primer on dinosaurs
DESCRIPTION
One of the best book on dinosaurs you will ever seeTRANSCRIPT
Lab Demo 0822 October 2015
PS4 Debrief• Precompute the MST of the original graph using
Prim or Kruskal
• Precompute all MiniMax to all vertices from the first ten vertices by running DFS/BFS
• Answer each query in O(1)
• Total time complexity O(E log V + 10 x V)
Intended solution for subtask D:
Build MST
Traverse from the first ten
vertices
PS4 Debrief• Precompute the MST of the original graph using
Prim or Kruskal
• Precompute all MiniMax to all vertices from the first ten vertices by running DFS/BFS
• Answer each query in O(1)
• Total time complexity O(E log V + 10 x V)
Intended solution for subtask D:
Why traversal only took O(V) time ?
PS4 Debrief
What if there is no constraint on the source vertex ?
PS4 Debrief1
2 3
10 4 6
5 7
9
8
Rooted tree at 1
PS4 Debrief1
2 3
10 4 6
5 7
9
8
Lowest Common Ancestor of 5 and 9
PS4 Debrief1
2 3
10 4 6
5 7
9
8
Lowest Common Ancestor (LCA) of 5 and 9
Ancestor of 5Ancestor of 9
PS4 Debrief
• Finding LCA of two vertices can be done in
• Finding maximum edge in path from a vertex to its ancestor can be done in
• Time complexity for each query becomes
• Total complexity O(E log V + Q log V)
O(log V)
O(log V)
O(log V)
PS5 is out :O
• Subtask C is quite tough to AC
• Need to use different codes for each subtask
• Try to start early
PS5 StatusName A B C D
- AC AC AC AC
Hubert AC AC AC
Rui Bin, Jiahao, Yun Shian, Danwen AC AC
Cindy, Bao Jun AC
The rest?
PS5• Given a graph, and Q queries
• Each query asks for shortest path from s to t that uses no more than k vertices
PS5 Subtask A• Given graph is an undirected weighted tree
• 1 ≤ V ≤ 1 000
• 1 ≤ Q ≤ 100 000
• k=V
PS5 Subtask A• Given graph is an undirected weighted tree
• 1 ≤ V ≤ 1 000
• 1 ≤ Q ≤ 100 000
• k=V Just ignore k
PS5 Subtask A• Given graph is an undirected weighted tree
• 1 ≤ V ≤ 1 000
• 1 ≤ Q ≤ 100 000
• k=V
Classic shortest path problem, use Dijkstra ?
Just ignore k
PS5 Subtask A• Given graph is an undirected weighted tree
• 1 ≤ V ≤ 1 000
• 1 ≤ Q ≤ 100 000
• k=V
Classic shortest path problem, use Dijkstra ?
Total complexity : O(Q * E log V) ~ 109 processes
Just ignore k
TLE👎
PS5 Subtask A• Given graph is an undirected weighted tree
• 1 ≤ V ≤ 1 000
• 1 ≤ Q ≤ 100 000
• k=V
Use property of a tree
Total complexity : O(Q * (V + E)) ~ 108 processes
run BFS/DFS from u to v
Just ignore k
TLE👎
PS5 Subtask A• Given graph is an undirected weighted tree
• 1 ≤ V ≤ 1 000
• 1 ≤ Q ≤ 100 000
• k=V
Precompute from each vertex
Total complexity : O(V * (V + E) + Q) ~ 106 processes
Notice the small V
AC👍
O(1) for each query
PS5 Subtask B• Given graph is a directed weighted graph
• 1 ≤ V ≤ 1 000
• 0 ≤ E ≤ 200 000
• 1 ≤ Q ≤ 10 000
• 0 ≤ s ≤ 9
• k=V
PS5 Subtask B• Given graph is a directed weighted graph
• 1 ≤ V ≤ 1 000
• 0 ≤ E ≤ 200 000
• 1 ≤ Q ≤ 10 000
• 0 ≤ s ≤ 9
• k=V
ignore k again
Shortest path on general graph
Dijkstra
Total complexity : O(Q * E log V) ~ 1010 processes
TLE👎
PS5 Subtask B• Given graph is a directed weighted graph
• 1 ≤ V ≤ 1 000
• 0 ≤ E ≤ 200 000
• 1 ≤ Q ≤ 10 000
• 0 ≤ s ≤ 9
• k=V
Shortest path on general graph
Dijkstra
Total complexity : O(Q * E log V) ~ 1010 processes
Looks like PS4 Subtask D
TLE👎
ignore k again
PS5 Subtask B• Given graph is a directed weighted graph
• 1 ≤ V ≤ 1 000
• 0 ≤ E ≤ 200 000
• 1 ≤ Q ≤ 10 000
• 0 ≤ s ≤ 9
• k=V
Precompute dijkstra from first ten vertices
Total complexity : O(10 * E log V + Q) ~ 2*107 processes
Looks like PS4 Subtask D
Query done in O(1)
AC👍
ignore k again
PS5 Subtask C• Given graph is a directed weighted graph
• 1 ≤ V ≤ 1 000
• 0 ≤ E ≤ 200 000
• 1 ≤ Q ≤ 20
• 1 ≤ k ≤ min(V, 20)
PS5 Subtask C• Given graph is a directed weighted graph
• 1 ≤ V ≤ 1 000
• 0 ≤ E ≤ 200 000
• 1 ≤ Q ≤ 20
• 1 ≤ k ≤ min(V, 20)
Need to use k now, cannot directly use Dijkstra
PS5 Subtask C• Given graph is a directed weighted graph
• 1 ≤ V ≤ 1 000
• 0 ≤ E ≤ 200 000
• 1 ≤ Q ≤ 20
• 1 ≤ k ≤ min(V, 20)
Need to somehow attach a counter to vertices
1
1,1 1,2 1,3 1,k-1 1,k…
Split each vertex to k new vertices
PS5 Subtask C• Given graph is a directed weighted graph
• 1 ≤ V ≤ 1 000
• 0 ≤ E ≤ 200 000
• 1 ≤ Q ≤ 20
• 1 ≤ k ≤ min(V, 20)
Need to somehow attach a counter to vertices
1
1,1 1,2 1,3 1,k-1 1,k…
Split each vertex to k new vertices
(u, k) moves to (v, k+1)
PS5 Subtask C0
1 2 3
Query from 0 to 3, when k = 3
PS5 Subtask C0
1 2 3
Query from 0 to 3, when k = 3
0,1 0,2 0,3
1,1
1,2
1,3
2,1
2,2
2,3
3,1
3,2
3,3
source
destination
Optimisation• Modified Dijkstra algorithm should run faster
• Constructing a new adjacency list with 20V vertices probably will get TLE
• Stop Dijkstra when it reach the destination vertex (important!)
• Use primitive int type on IntegerPair object
Subtask C has a very strict time limit
AC👌
PS5 Subtask D
• Subtask D is beyond CS2010
• It is a classical problem that can be solved using maximum flow.
• Take CS3233 next sem if you are interested.
UVa 926
• This is part of CS2010 final exam question in the past 4 years
UVa 926• Given a grid, start position
and end position
• Find number of ways to go from start to end
• Cannot move south or west.
• Some roads are discontinued. (1,2) to (2,2) and (2,2) to (2,3)
UVa 926• Represent each intersection as a vertex
• There are no cycle in this graph, so this is a DAG (Directed Acyclic Graph)
UVa 926• Counting number of path is usually a DP problem.
• Since it is a DAG, so it is okay to use DP.
Let start position be (xs, ys)
Let end position be (xe, ye)
DP solutioncountPath(x, y): number of path from (x,y) to (xe, ye)
countPath(x,y) = 1
if (x,y) = (xe, ye) Base Case
DP solutioncountPath(x, y): number of path from (x,y) to (xe, ye)
countPath(x,y) =
if (x,y) <> (xe, ye)
countPath(x+1, y) countPath(x, y+1)
0 0
if (x,y) to (x+1,y) not under construction
+
otherwise otherwise
if (x,y) to (x,y+1) not under construction
