graphs, bfs, dfs and more… ctldenon. agenda review – graphs – graphs representations – dfs...

Graphs, BFS, DFS and More…

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Agenda

• Review– Graphs – Graphs Representations– DFS– BFS

• More– Topological-Sort– Shortest Path

• Dijkstra’s , Bellman-Ford

– All Pairs Shortest Path• Dijkstra’s, Floyd-WarShall

Graphs

Graphs (review)• Definition. A directed graph (digraph)

– G = (V, E) is an ordered pair consisting of– a set V of vertices (singular: vertex)– a set E V × V of edges⊆

In an undirected graph G = (V, E), the edge set E consists of unordered pairs of vertices.

More,• Connected, Disconnected• Tree, Forest• Cyclic, Acyclic• Weighted, un-weighted• Multigraph …

Graphs Representations - adjacency matrix

Graphs Representations - adjacency matrix

The adjacency matrix of a graph G = (V, E), where V = {1, 2, …, n}, is the matrix A[1 . . n, 1 . . n] given by

A[i, j] = 1 if (i, j) E, (What if multigraph ???)∈0 if (i, j) E.∉

Storage size??

Graphs Representations - adjacency list

• An adjacency list of a vertex v V is the list ∈Adj[v] of vertices adjacent to v.

• Storage size??

DFS and BFS

• Review• Possible

ordering of traversing if DFS?

• Possible ordering of traversing if BFS?

BFS

BFS(G)marks all Vertices in G as un-visitedmarks start vertex s as visitedenqueue s into queue Q

while queue not emptydequeue the first vertex u from queue Qfor each vertex v directly reachable from u

if v is unvisitedenqueue v to queue Qmark v as visited

BFS

• Application– Shortest Path of un-weighted

while queue not emptydequeue the first vertex u from queuefor each neighbor vertex v of u

if v is unvisitedenqueue v to queuemark v as visitedparent[v] = u

DFSDFS-visit(u)

color[u] = GRAYfor each neighbor vertex v of u

if color[v] = WHITEDFS-visit[v]

DFS(G)for each vertex u in G

color[u] = WHITE

for each vertex u in Gif color[u] = WHITE

DFS-visit(u)

Topological Sort

Linear ordering of a directed acyclic graph (DAG)’ s nodes in which each node comes before all nodes to which it has outbound edges.

http://en.wikipedia.org/wiki/Directed_acyclic_graph

http://en.wikipedia.org/wiki/Directed_acyclic_graph

Topological SortDFS-visit(u)

color[u] = GRAYfor each neighbor vertex v of u

if color[v] = WHITEparent[v] = uDFS-visit[v]

color[u] = BLACKtime = time + 1f[u] = time

DFS(G)for each vertex u in G

color[u] = WHITEparent[u] = Nil

time = 0

for each vertex u in Gif color[u] = WHITE

DFS-visit(u)

Topological Sort

• Call DFS(G) to compute the finishing times f[v] for each vertex v

• As each vertex is finished (Mark as Black), insert it onto the front of a list

• Return the list

Shortest Path

Shortest Path

• More Formal Definition– Consider a digraph G = (V, E) with edge-weight function w : E

→ R. The weight of path p = v1 → v2 → … → vk is defined to be

• A shortest path from u to v is a path of minimum weight from u to v. The shortest-path weight from u to v is defined as (u, v) = min{w(p) : p is a path from u to v}.

• Note: δ(u, v) = ∞ if no path from u to v exists.

Shortest Path

• Optimal substructure– Theorem. A subpath of a shortest path is a

shortest path.

Shortest Path

• Triangle inequality– Theorem. For all u, v, x V, we have δ(u, v) ≤δ(u, ∈

x) + δ(x, v).

– Relaxation: if d[v] > d[x] + w(x, v)then d[v] ← d[x] + w(x, v)

Shortest Path

• Un-weighted graph?

Shortest Path

• Un-weighted graph? BFS

Shortest Path

• Un-weighted graph? BFS• Non-negative edge weights?

Shortest Path

• Un-weighted graph? BFS• Non-negative edge weights? Dijkstra’s

algorithm

Shortest Path – Dijkstra’s algorithmSuppose you create a knotted web of strings, with each knot corresponding to a node, and the strings corresponding to the edges of the web: the length of each string is proportional to the weight of each edge. Now you compress the web into a small pile without making any knots or tangles in it. You then grab your starting knot and pull straight up. As new knots start to come up with the original, you can measure the straight up-down distance to these knots: this must be the shortest distance from the starting node to the destination node. The acts of "pulling up" and "measuring" must be abstracted for the computer, but the general idea of the algorithm is the same: you have two sets, one of knots that are on the table, and another of knots that are in the air. Every step of the algorithm, you take the closest knot from the table and pull it into the air, and mark it with its length. If any knots are left on the table when you're done, you mark them with the distance infinity.

Shortest Path – Dijkstra’s algorithm

Shortest Path – Dijkstra’s algorithmvoid dijkstra(int s){ memset(visit,0,sizeof(visit)); memset(dist,0x7f,sizeof(dist)); priority_queue<C> pq; pq.push(C(s,0)); dist[s] = 0; while(pq.size()>0){ C c = pq.top(); pq.pop(); int u = c.u; if(visit[u]) continue; visit[u] = true; for(int i=0;i<adjc[u];++i){ int v = adj[u][i]; if(visit[v]) continue; if(c.c+cost[u][v]<dist[v]){ dist[v] = dist[u] + cost[u][v]; pred[v] = u; pq.push(C(v, dist[v])); } } } }

int dist[N]; int pred[N]; struct C{ int u,c; C(){} C(int a,int b):u(a),c(b){} bool operator<(const C&a) const { return c > a.c; } }; // the graph is represented by // a adjacency list // a vertex a has adjc[a] neighbors // adj[a][0] … adj[a][adjc[a]-1] are // neighbors of vertex // cost[a][b] has edge cost for (a,b) int adjc[N]; int adj[N][N]; int cost[N][N]; bool visit[N];

Shortest Path


algorithm

Shortest Path


algorithm• Negative edge weights?

Shortest Path


algorithm• Negative edge weights? Bellman-Ford

algorithm

Shortest Path – Bellman Ford

• If a graph G = (V, E) contains a negative-• weight cycle, then some shortest paths may

not exist.

• Bellman-Ford algorithm:– Finds all shortest-path lengths from a sources V ∈

to all v V or determines that a negative-weight ∈cycle exists.



• Theorem.– If G= (V, E)contains no negative-weight cycles, then after the

Bellman-Ford algorithm executes, d[v] = δ(s, v)for all v V. ∈

• Proof. – Let v V be any vertex, and consider a shortest path p from s ∈

to v with the minimum number of edges.

– Since p is a shortest path, we have δ(s, vi) = δ(s, vi–1) + w(vi–1, vi) .


If G contains no negative-weight cycles, p is simple. Longest simple path has ≤|V|–1edges.

Initially, d[v0] = 0 = δ(s, v0), and d[v0] is unchanged by subsequent relaxations (because of the lemma from Lecture 14 that d[v] ≥δ(s, v)).

• After 1pass through E, we have d[v1] = δ(s, v1).• After 2passes through E, we have d[v2] = δ(s, v2).M• After k - passes through E, we have d[vk] = δ(s, vk).


• If a value d[v] fails to converge (stop changing) after |V|–1passes, there exists a negative-weight cycle in G reachable from s

Shortest Path



algorithm• DAG?

Shortest Path



algorithm• DAG? Topological Sort + One Round of

Bellman-Ford

Shortest Path



algorithm• DAG? Topological Sort + One Round of

Bellman-Ford• All Pairs Shortest Path?

All Pairs Shortest Path

• Non-negative edge weights


• Non-negative edge weights– Dijkstra’s algorithm |V| times



• Negative edge



• Negative edge– Floyd-WarShall (Not the fastest one, but good

enough in most cases)

All Pairs Shortest Path - Floyd-WarShall

• Define Cij(k)=weight of a shortest path from I to j with intermediate vertices belonging to the set {1, 2, …, k}.

• Thus, δ(i, j) = Cij(n). Also, Cij(0) = w(i, j) .


Cij(k)= min { Cij(k–1),Cik(k–1)+ C kj(k–1) }

intermediate vertices in {1, 2, …, k}

Reference

• Depth-First Search http://en.wikipedia.org/wiki/Depth-first_search

• Graph Algorithms http://i.cs.hku.hk/~provinci/training07/02-Graph_Theory.ppt

• Topological Sorting http://en.wikipedia.org/wiki/Topological_sorting

• Greedy Algorithms (and Graphs) http://velblod.videolectures.net/2005/ocw/mit/6.046j/mit6046jf05_leiserson_lec16/mit6046jf05_leiserson_lec16_01.pdf

• Shortest Paths I http://carbon.videolectures.net/2005/ocw/mit/6.046j/mit6046jf05_demaine_lec17/mit6046jf05_demaine_lec17_01.pdf

• Shortest Paths II http://carbon.videolectures.net/2005/ocw/mit/6.046j/mit6046jf05_demaine_lec18/mit6046jf05_demaine_lec18_01.pdf

• Shortest Paths III http://carbon.videolectures.net/2005/ocw/mit/6.046j/mit6046jf05_demaine_lec18/mit6046jf05_demaine_lec19_01.pdf

http://en.wikipedia.org/wiki/Depth-first_search

http://i.cs.hku.hk/~provinci/training07/02-Graph_Theory.ppt

http://en.wikipedia.org/wiki/Topological_sorting

http://en.wikipedia.org/wiki/Topological_sorting

http://velblod.videolectures.net/2005/ocw/mit/6.046j/mit6046jf05_leiserson_lec16/mit6046jf05_leiserson_lec16_01.pdf



http://carbon.videolectures.net/2005/ocw/mit/6.046j/mit6046jf05_demaine_lec17/mit6046jf05_demaine_lec17_01.pdf







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