Evaluation of the Course (Modified) Course work: 30% Four assignments (25%) 7.5 5 points for each of the first two three assignments 10 points for the last assignment One term paper (5%) (week13 Friday) Find an open problem from internet. State the problem definition in English. Write the definition mathematically. Summarize the current status No more than 1 page A final exam: 70% 2019/4/11 CS4335 Design and Analysis of Algorithms /Shuai Cheng Li

Single source shortest path with negative cost edges 2019/4/11 chapter25

Shortest Paths: Dynamic Programming Def. OPT(i, v)=length of shortest s-v path P using at most i edges. Case 1: P uses at most i-1 edges. OPT(i, v) = OPT(i-1, v) Case 2: P uses exactly i edges. If (w, v) is the last edge, then OPT use the best s-w path using at most i-1 edges and edge (w, v). Remark: if no negative cycles, then OPT(n-1, v)=length of shortest s-v path.  Cwv s w v 2019/4/11 chapter25 OPT(0, s)=0.

Shortest Paths: implementation Shortest-Path(G, t) { for each node v  V M[0, v] =  M[0, s] = 0 for i = 1 to n-1 for each node w  V M[i, w] = M[i-1, w] for each edge (w, v)  E M[i, v] = min { M[i, v], M[i-1, w] + cwv } } Analysis. O(mn) time, O(n2) space. m--no. of edges, n—no. of nodes Finding the shortest paths. Maintain a "successor" for each table entry. 2019/4/11 chapter25

Shortest Paths: Practical implementations Practical improvements. Maintain only one array M[v] = shortest v-t path that we have found so far. No need to check edges of the form (w, v) unless M[w] changed in previous iteration. Theorem. Throughout the algorithm, M[v] is the length of some s-v path, and after i rounds of updates, the value M[v]  the length of shortest s-v path using  i edges. Overall impact. Memory: O(m + n). Running time: O(mn) worst case, but substantially faster in practice. 2019/4/11 chapter25

Bellman-Ford: Efficient Implementation Push-Based-Shortest-Path(G, s, t) { for each node v  V { M[v] =  successor[v] = empty } M[s] = 0 for i = 1 to n-1 { for each node w  V { if (M[w] has been updated in previous iteration) { for each node v such that (w, v)  E { if (M[v] > M[w] + cwv) { M[v] = M[w] + cwv successor[v] = w If no M[w] value changed in iteration i, stop. Note: Dijkstra’s Algorithm select a w with the smallest M[w] . Time O(mn), space O(n). 2019/4/11 chapter25

6 7 9 2 5 -2 8 -3 -4 s u v x y (a) 2019/4/11 chapter25

u 5 v 6 8 -2 6 -3 8 7 s -4 2 7 7 8 9 x y (b) 2019/4/11 chapter25

u 5 v 6 4 -2 6 -3 8 7 s -4 2 7 7 2 9 x y (c) 2019/4/11 chapter25

u 5 v 2 4 -2 6 -3 8 7 s -4 2 7 7 2 9 x y (d) 2019/4/11 chapter25

u 5 v 2 4 -2 6 -3 8 7 s -4 2 7 7 -2 y x (e) 9 vertex: s u v x y 7 s -4 2 7 7 -2 y x (e) 9 vertex: s u v x y d: 0 2 4 7 -2 successor: s v x s u 2019/4/11 chapter25

Corollary: If negative-weight circuit exists in the given graph, in the n-th iteration, the cost of a shortest path from s to some node v will be further reduced. Demonstrated by the following example. 2019/4/11 chapter25

An example with negative-weight cycle 5  1 6 -2 8 7 7 9 2 2 5 -8 An example with negative-weight cycle 2019/4/11 chapter25

7 6  8 5 -2 1 2 9 -8 i=1 2019/4/11 chapter25

7 6 16  11 9 8 5 -2 1 2 -8 i=2 2019/4/11 chapter25

7 6 16 12 11 1 9 8 5 -2 2 -8 i=3 2019/4/11 chapter25

6 16 12 11 1 9 7 8 5 -2 2 -8 i=4 2019/4/11 chapter25

5 6 11 1 6 -2 8 7 12 7 9 2 6 15 2 5 -8 8 1 i=5 2019/4/11 chapter25

6 15 12 11 8 7 5 -2 1 2 9 -8 i=6 2019/4/11 chapter25

5 6 11 1 6 -2 8 7 12 7 9 2 5 15 2 5 -8 8 x i=7 2019/4/11 chapter25

5 6 11 1 6 -2 8 7 12 7 9 2 5 15 2 5 -8 7 x i=8 2019/4/11 chapter25

Dijkstra’s Algorithm: (Recall) Dijkstra’s algorithm assumes that w(e)0 for each e in the graph. maintain a set S of vertices such that Every vertex v S, d[v]=(s, v), i.e., the shortest-path from s to v has been found. (Intial values: S=empty, d[s]=0 and d[v]=) (a) select the vertex uV-S such that d[u]=min {d[x]|x V-S}. Set S=S{u} (b) for each node v adjacent to u do RELAX(u, v, w). Repeat step (a) and (b) until S=V. 2019/4/11 chapter25

Continue: DIJKSTRA(G,w,s): INITIALIZE-SINGLE-SOURCE(G,s) S Q V[G] while Q do u EXTRACT -MIN(Q) S S {u} for each vertex v  Adj[u] do RELAX(u,v,w) 2019/4/11 chapter25

u v 10 5 2 1 3 4 6 9 7 8 s Single Source Shortest Path Problem x y (a) 2019/4/11 chapter25

u v 1 10/s 8 10 9 s 2 3 4 6 7 5 5/s 8 2 x y (b) (s,x) is the shortest path using one edge. It is also the shortest path from s to x. 2019/4/11 chapter25

u v 1 8/x 14/x 10 9 s 2 3 4 6 7 5 5/s 7/x 2 x y (c) 2019/4/11 chapter25

u v 1 8/x 13/y 10 9 s 2 3 4 6 7 5 5/s 7/x 2 x y (d) 2019/4/11 chapter25

7/x 9/u 5/s 8/x 10 5 2 1 3 4 6 9 7 s u v x y (e) 2019/4/11 chapter25

Backtracking: v-u-x-s 1 8/x 9/u 10 9 s 2 3 4 6 7 5 5/s 7/x 2 x y (f) Backtracking: v-u-x-s 2019/4/11 chapter25

Proof: We prove it by induction on |S|. Theorem: Consider the set S at any time in the algorithm’s execution. For each vS, the path Pv is a shortest s-v path. Proof: We prove it by induction on |S|. If |S|=1, then the theorem holds. (Because d[s]=0 and S={s}.) Suppose htat the theorem is true for |S|=k for some k>0. Now, we grow S to size k+1 by adding the node v. 2019/4/11 chapter25

Proof: (continue) Now, we grow S to size k+1 by adding the node v. Let (u, v) be the last edge on our s-v path Pv. Consider any other path from P: s,…,x,y, …, v. (red in the Fig.) y is the first node that is not in S and xS. Since we always select the node with the smallest value d[] in the algorithm, we have d[v]d[y]. Moreover, the length of each edge is 0. Thus, the length of Pd[y]d[v]. That is, the length of any path d[v]. Therefore, our path Pv is the shortest. y x s If y does not exist, d[v] is the smallest length for paths from s to v using red nodes only since we did relax.from every red node to v. u v Set S 2019/4/11 chapter25

S->v is shorter than s->u, but it is longer than The algorithm does not work if there are negative weight edges in the graph . u -10 2 v s 1 S->v is shorter than s->u, but it is longer than s->u->v. 2019/4/11 chapter25

0-1 version v/w: 1, 3, 3.6, 3.66, 4. 2019/4/11 chapter25

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