Lecture I: Shortest Path Algorithms

Size: px

Start display at page:

Download "Lecture I: Shortest Path Algorithms"

Aileen Lee
5 years ago
Views:

1 Lecture I: Shortest Path Algorithms Dr Kieran T. Herley Department of Computer Science University College Cork October 201 KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

2 Background Shortest Path Algorithms Setting: directed graph, real edge weights Let the length of a path be the sum of its edge weights and let δ(s, u) = length of shortest path from s to u Reduces to edge distance if all weights are 1 KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

3 Background Example Single source shortest path problem Given a weighted graph G and a designated node s G, determine δ(s, u) for each u G. u δ(s, u) s 0 a 8 b 9 c 7 d 5 All pairs shortest path problem Given a weighted graph G, determine δ(x, y) for each x, y G. KH (21/10/1) Lecture I: Shortest Path Algorithms October 201 / 28

4 Floyd-Warshall Algorithm Graphs and Matrices A Graph and its matrix representation (nodes listed alpahetically a = 1 etc.) if i = j w i,j = weight(i, j) if i j and (i, j) E if i j and (i, j) E No negative cycles allowed otherwise shortest path notion not well defined KH (21/10/1) Lecture I: Shortest Path Algorithms October 201 / 28

Floyd-Warshall Algorithm Some Terminology k-path path all of whose intermediate nodes are numbered less than or equal to k; (the start/end nodes may have numbers greater than k).

5 Floyd-Warshall Algorithm Some Terminology k-path path all of whose intermediate nodes are numbered less than or equal to k; (the start/end nodes may have numbers greater than k). Consider shortest k-path from i to j : Best path is π is better of α or β k γ either it passes through node k (once): 2- i.e. consists of β = shortest (k 1)-path from i to k node k γ = shortest (k 1)-path from k to j or it does not pass through node k: α = shortest (k 1)-path from i to j KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

6 Floyd-Warshall Algorithm A useful identity Definition Claim l (k) i,j = length of shortest k-path joining i to j. l (k) i,j = { wi,j if k = 0 min{l (k 1) i,j, l (k 1) i,k + l (k 1) k,j } if k > 0 KH (21/10/1) Lecture I: Shortest Path Algorithms October 201 / 28

7 Floyd-Warshall Algorithm Observation Visualize the set of l (k) i,j quantities as three-dimensional grid with point l (k) i,j having co-ordinates (i, j, k). l (k) i,j = { wi,j if k = 0 min{l (k 1) i,j, l (k 1) i,k + l (k 1) k,j } if k > 0 Notice: each horizontal slice represents values l (k) i,j for some fixed k the quantities in the bottom slice (k = 0) depend only on edge weights KH (21/10/1) each quantity l (k) Lecture on slice I: Shortest k depends Path Algorithms only on quantities October on the201 slice 7 / 28

8 Floyd-Warshall Algorithm Floyd-Warshall Algorithm Algorithm FLOYD WARSHALL(G): W = matrix of edge weights for i 1 to n do for j 1 to n do L[ i, j, 0] W[i, j ] for k 1 to n do for i 1 to n do for j 1 to n do if L[ i, j, k 1] < L[i, k, k 1] + L[k, j, k 1] L[ i, j, k] L[i, j, k 1] else L[ i, j, k] L[i, k, k 1] + L[k, j, k 1] KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

9 Floyd-Warshall Algorithm Notes Length of shortest i-to-j path in L[i, j, n]. Negative weight OK, but not negative cycles. Running time: O(n ) Can get by with 2D arrays (two for alternate slices) Can be modified to produce paths (not just lengths) KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

10 Floyd-Warshall Algorithm Transitive Closure The transitive closure of G = (V, E) the the graph G = (v, E ) such that there is an edge in G from u ti v if and only if there is a path in G from u to v. Algorithm: Give weight 1 to each edge in G; Run Floyd-Warshall algorithm; Each L[i, j, n] denotes path in G, so add corresponding edge (i, j) to E. KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

11 Single-Source Shortest Path Problem Single source shortest path problem: Given a weighted graph G and a designated node s G, determine δ(s, u) for each u G. Disallow negative edges Could use Floyd-Warshall, but is there a more efficient way? KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

12 Single-source shortest path problem (SSSP) s Problem t y 7 9 x z Path sequence of edge-connected vertices Path length sum of edge lengths Shortest path path between endpoints with minimum total length (SSSP) Given graph G and source vertex s, calculate length of shortest path from source s to each vertex in G. KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

13 Dijkstra s Algorithm s t x s 0 t x 8 Inputs (left): y z G: graph with nonnegative (important!) edges s: start node Outputs (right): Compute for each node v, δ(s, v) = length shortest path from s to v ( if none). Shown inside nodes. y 7 z KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

14 Idea Begin with a crude estimate d[u] for δ(s, u): { 0 if u = s d[u] = otherwise Refine estimates using (carefully chosen) sequence of the following edge operations (i.e. (u, v) must be an edge): Algorithm Relax(u, v): if d[v] > d[u] + weight(u, v) then d[v] d[u] + weight(u, v) Ultimately (we hope) d[u] = δ(s, u), for all u. KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

15 Notes Observation Through any sequence of Relax(u, v) steps d[x] is non-increasing, and If d[x], then there is a path of length d[x] in G from s to x. Algorithm Relax(u, v): if d[v] > d[u] + weight(u, v) then d[v] d[u] + weight(u, v) KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

16 Dijkstra s Algorithm Algorithm Relax(u, v): if d[v] > d[u] + weight(u, v) then d[v] d[u] + weight(u, v) Algorithm Dijkstra(G, s ): Initialize d(s) to 0 Initialize d(x) = INFTY for all x s Create priority queue Q and place each node x in Q with key value d(x) while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

17 Priority queue refresher What Container abstraction holding key, value pairs Operations insert Add new key value item to container min Return the value with the smallest key (ties broken arb.) remove min Return and remove the value with the smallest key Implementation Heap-ordered tree stored in array; insert, remove min O(log n) time. KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

18 D s algorithm in action s 0 t x Q vertex non-q vertex Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) y z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

19 D s algorithm in action s 0 t x Q vertex non-q vertex Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) y z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

20 D s algorithm in action s 0 t x Q vertex non-q vertex Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) y z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

21 D s algorithm in action s 0 1 t x Q vertex non-q vertex y z u vertex Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

22 D s algorithm in action s 0 t x Q vertex non-q vertex Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) y z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

23 D s algorithm in action s 0 t x Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

24 D s algorithm in action s 0 1 t x Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

25 D s algorithm in action s 0 t x Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

26 D s algorithm in action s 0 t x 1 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

27 D s algorithm in action s 0 t x 1 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

28 D s algorithm in action s 0 t x 1 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

29 D s algorithm in action s 0 t x 8 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

30 D s algorithm in action s 0 t x 8 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

31 D s algorithm in action s 0 t x 8 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

32 D s algorithm in action s 0 t x 8 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

33 D s algorithm in action s 0 t x 8 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

34 D s algorithm in action s 0 t x 8 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

35 D s algorithm in action s 0 t x 8 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

36 D s algorithm in action s 0 t x 8 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

37 D s algorithm in action s 0 t x 8 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

38 D s algorithm in action s 0 t x 8 Q vertex non-q vertex y Algorithm Dijkstra(G, s ):... while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) 7 z u vertex KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

39 Aside Relax involves changes to d[] i.e. priority queue keys. New priority queue operation: reduce key(x, k) Replace key of item x with smaller key k Implementation note: Effectively helper method heap decrease key used in insert operation Running time: O(log(queue size)) KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

40 Running time of D s algorithm Algorithm Relax(u, v): if d[v] > d[u] + weight(u, v) then d[v] d[u] + weight(u, v) Algorithm Dijkstra(G, s ): / Initialization stuff / while Q is not empty do u Q.remove min element() for each v in G.neighbours(u) do Relax(u, v) n, m = num. nodes, edges Priority queue operations: O((n + m) log n) (over) Non-priority queue stuff: O(n + m) Total running time: O((n + m) log n) KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

41 Notes Priority queue operations op no. cost per subtotal insert n O(log n) O(n log n) isempty n O(1) O(n) remove min element n O(log n) O(n log n) replacekey m O(log n) O(m log n) Non-priority queue stuff n iterations in all for fixed u, for-loop takes O(#edges leaving u) time Total O(n) + O(#edges leaving u) = O(n + m) }{{} u V while } {{ } for O((n + m) log n) + O(n + m) = O((n + m) log n) KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

42 Claim 1 Definition We say x is settled once d(x) = δ(s, x). Claim If u is settled and s u v is a shortest path to v, then following Relax(u, v) v is also settled KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

43 Claim 1 Definition We say x is settled once d(x) = δ(s, x). Claim If u is settled and s u v is a shortest path to v, then following Relax(u, v) v is also settled Before: Relax(u, v), d(u) = δ(s, u) After: Relax(u, v), d(v) d(u) + w(u, v) KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

44 Claim 1 Definition We say x is settled once d(x) = δ(s, x). Claim If u is settled and s u v is a shortest path to v, then following Relax(u, v) v is also settled Before: Relax(u, v), d(u) = δ(s, u) After: Relax(u, v), d(v) d(u) + w(u, v) This must equal δ(s, v): shortest path assumption KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

45 Claim 2 Claim Node u is settled before being removed from Q. Proof. Suppose this were not true Let u be the first node removed from Q that was not settled Let y be first node along shortest path from s to u that is in Q and x be its predecessor KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

46 Claim 2 cont d Claim Node u is settled before being removed from Q. d(u) d(y) by EXTRACTMINs choice of u = δ(s, y) x not in Q so d(x) = δ(s, x); Relax(x, y) operation ensures that d(y) = δ(s, y) δ(s, u) since s-to-y a prefix of s-to-u and edge-weights are non-negative d(u) KH (21/10/1) Lecture I: Shortest Path Algorithms Hence d(u) = δ(s, October u) / 28

47 Constructing the Paths Minor tweak to Relax to record shortest-path predecessors so algorithms constructs paths not just their length. Algorithm Relax(u, v): if d[v] > d[u] + weight(u, v) then d[v] d[u] + weight(u, v) p[v] u At conclusion, p(x) indicates parent of x in a (shortest-path) tree rooted at s. The root to leaf path to x is a shortest path. KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

48 SP Algorithms and Network Routing network of hosts/switches(nodes) and links (edges) edge weight desirability of using edge Determine for/at each node, the lowest cost route to each destination Would like approach to be distributed resilient to failures/changes KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

49 Idea 1 Each node determines its neighbours/edges and costs and injects into into network. (Re-injects following changes) Nodes exchange this information among themselves Each node gets complete map of the network Changes get reflected in maps as new info replaces old KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

50 Idea 2 Each node x runs Dijkstra locally to establish best paths from x to other network nodes Maintains table of (dest., cost, next-hop) entries for routing decisions Information re-computed when network map changes KH (21/10/1) Lecture I: Shortest Path Algorithms October / 28

Lecture 11: Analysis of Algorithms (CS ) 1

Lecture 11: Analysis of Algorithms (CS583-002) 1 Amarda Shehu November 12, 2014 1 Some material adapted from Kevin Wayne s Algorithm Class @ Princeton 1 2 Dynamic Programming Approach Floyd-Warshall Shortest