Shortest Paths 08-07-2006 PowerPoint adapted from Alan Tam’sShortest Path, 2004 with slight modifications
Breadth First Search (BFS) • Techniques to Enumerate all Vertices for Goal • BFS needs O(V) queue and O(V) set for duplicate elimination and runs in O(V + E) time • BFS can find Shortest Path if Graph is Not Weighted • BFS works because a Queue ensures a specific Order
What Order? • Define d[v] be the length of shortest path from s to v • At any time, vertices are classified into: • Black With known d[v] • Grey With some known path • White Not yet touched • The Grey vertex with shortest known path has potential to become Black • The Head of Queue is always the case in an unweighted graph • Why?
Weighted Graph • Queue does not promise smallest d[v] anymore • Expanding weighted edge causeds unnecessary searching of artificial vertices • We can simply pick the real vertex nearest to the starting vertex • We need “Sorted Queue” which “dequeues” vertices in increasing order of d[v]. • It is called a “Priority Queue” and negation of d[v] is called the Priority of vertex v
0 3 2 1 3 3 3 1
0 3 3 2 1 3 3 2 3 1 3
Done? 0 3 3 2 1 3 5 3 2 3 1 3
Done? 0 3 3 2 1 3 4 3 2 3 1 3
0 3 3 2 1 3 4 3 2 3 1 3
Done? 0 3 3 2 1 3 4 3 2 3 1 3
Dijkstra’s Algorithm d ← ∞ d[s] ← 0 enqueue(queue, s) whilenot empty(queue) do u = dequeue(queue) for-eachvwhere (u, v) in E if d[v] > d[u] + wuvthen d[v] = d[u] + wuv enqueue(queue, v) • Note that d[v] represents the known shortest distance of v from s during the process and may not be the real shortest distance until v is marked black.
Requirements for Priority Queue • Map<V, int> • Store temporary values of d[v] • PriorityQueue<V, function: V→int > • Extract vertices in increasing order of d[v] • Initially all d[v] are infinity • When we visit a White vertex • Set d[v] • Insert v into Queue • When we visit a Gray vertex • Update d[v] • What to do to the Priority Queue?
Usage: Insert: V Extract: V Decrease: E - V Array Insert: O(1) Extract: O(n) Decrease: O(1) Sorted Array Insert: O(n) Extract: O(1) Decrease: O(n) Heap Insert: O(lg n) Extract: O(lg n) Decrease: O(lg n) Fibonacci Heap Insert: O(1) Extract: Amortized O(lg n) Decrease: O(1) Implementing Priority Queue?
Time Complexity
Time Complexity
Another Attempt • Consider the following code segment: For each edge (u, v) If d[v] > d[u] + wuv d[v] ← d[u] + wuv • Assume one of the shortest paths is (s, v1, v2, …, vk) • If d[vi] = its shortest path from s • After this loop, d[vi+1] = its shortest path from s • By MI, After k such loops, found shortest path from s to vk
Bellman-Ford Algorithm • All v1, v2, …,vk distinct? Do V-1 times for-each (u, v) in E if d[v] > d[u] + wuv then d[v] ← d[u] + wuv • Order = O(VE) • Support Negative-weight Edges
Another Attempt • Number vertices as v1, v2, …, vV • Define no≤k-path as a path which does not pass through vertices v1, v2, …, vk • A path s-t must either be • a no≤1-path, or • concatenation of a no≤1-path s-v1 and a no≤1-path v1-t • A no≤1-path s-t must either be • a no≤2-path, or • concatenation of a no≤2-path s-v2 and a no≤2-path v2-t • By MI …
For all S For all T S T Trivial Paths Direct Path from S to T
For all R For all S Direct Path from R to S R S Direct Path from R to T For all S For all T Direct Path from S to T S T For S = vV
For all S For all T S T no≤(V-1)-Paths Shortest Path from S to Tvia vV
For all R For all S Shortest Path from R to Svia vV R S Shortest Path from R to Tvia vV For all S For all T Shortest Path from S to Tvia vV S T For S = vV-1
For all S For all T S T no≤(V-2)-Paths Shortest Path from S to Tvia vV-1,vV
For all S For all T S T no≤2-Paths Shortest Path from S to Tvia v3,v4,...,vV
For all R For all S Shortest Path from R to Svia v3,v4,...,vV R S Shortest Path from R to Tvia v3,v4,...vV For all S For all T Shortest Path from S to Tvia v3,v4,...,vV S T For S = V2
For all S For all T S T no≤1-Paths Shortest Path from S to Tvia v2,v3,...,vV
For all R For all S Shortest Path from R to Svia v2,v3,...,vV R S Shortest Path from R to Tvia v2,v3,...vV For all S For all T Shortest Path from S to Tvia v2,v3,...,vV S T For S = V1
For all S For all T S T Shortest Paths Shortest Path from S to Tvia v1,v2,...,vV
Warshall-Floyd Algorithm for-each (u, v) in E d[u][v] ← wuv for-eachi in V for-eachjin V for-eachkin V if d[j][k] > d[j][i] + d[i][k] d[j][k] ← d[j][i] + d[i][k] • Time Complexity: O(V3)
Graph Modeling • Conversion of a problem into a graph problem • Sometimes a problem can be easily solved once its underlying graph model is recognized • Graph modeling appears almost every year in NOI or IOI (cx, 2004)
Basics of Graph Modeling • A few steps: • identify the vertices and the edges • identify the objective of the problem • state the objective in graph terms • implementation: • construct the graph from the input instance • run the suitable graph algorithms on the graph • convert the output to the required format (cx, 2004)
