1. CSE 202 - Shortest Paths1 Midterm page 1 The worst-case complexity of an algorithm is... a function from N (or size of instance) to N.  (n lg n) is... a set of functions. A random variable is... a function from elementary event to real number. Polyhash is... a set of (hash) functions. T(n) = 3 T(n/2) + n T(n) is  (n lg3 ) T(n) = 3 T(n/3) + 5 T(n) is  (n) T(n) = 3 T(n/4) + n lg n T(n) is  (n lg n) False A heap of height n can have 2 n+1 –1 nodes False A tree of height k might only have k+1 nodes True A red-black can implement a priority queue efficiently

2. CSE 202 - Shortest Paths2 Midterm page 2 Yes, you can implement “Decrease-Key(S,x,k)” in a max heap efficiently. Go to node x (which you can do since you have a pointer to it), change its priority to k, apply Heapify (which “bubbles downward”). Given a red-black tree... in black subtree –Every non-leaf has 2, 3, or 4 children (by P3). –Every leaf has depth k (by P4). –Thus # black nodes  2 k+1 – 1 (obvious, or by induction) In full red-black tree, N  # black nodes  2 k+1 – 1 –So lg N  lg (2 k+1 – 1)  lg 2 k = k; that is, k < lg N. –Longest path  2 k (by P4). –So height tree = longest path  2 lg N. NOTE: Just saying “Tree is balanced” is insufficient – we never even really defined what “balanced” means.

3. CSE 202 - Shortest Paths3 Midterm page 3 If keys are more or less uniformly randomly distributed (so Bucket Sort take O(N) time) and K is large (or grows with N, so KN isn’t O(N)), then Bucket Sort will be faster than Radix Sort. If the keys are very skewed, and K is small, Radix Sort will be faster (actually, all you really need is that K is very small, that is, K << lg N). Greedy change-making can be arbitrarily bad. Given any c, let D 2 = 2c, D 3 = 2c+1, and N = 4c. Then greedy will give one D 3 and 2c-1 D 1 ’s, or 2c coins. But optimal is to give two D 2 ’s, or 2 coins. So greedy is c times worse.

4. CSE 202 - Shortest Paths4 Midterm page 5 Let A(i) = best sum of numbers in the first i columns that doesn’t use any entry in column i. B(i) = best sum [of...] that uses only the top entry in column i C(i) = best sum that uses middle entry in column i D(i) = best sum that uses only the bottom entry in column i E(i) = best sum that uses top and bottom entry of column i. I claim, given A(i),..., E(i), we can compute A(i+1),..., E(i+1) in the “obvious” way. This will take  (N) time. E.g. A(i+1) = max(A(i), B(i), C(i), D(i), E(i)) B(i+1) = T(1,i+1) + max(A(i), C(i), D(i)), etc.

5. CSE 202 - Shortest Paths5 Midterm, page 6 Basic idea: use divide an conquer. ktile(A,N,K) { /* print the K-tiles of A[1],..., A[N] */ Let M = select(A, N, K/2); /* takes O(N) time */ Split A by M into Big and Small (each of size N/2); if (K>1) call ktile (Big, N/2, K/2); print M; if (K>1) call ktile (Small, N/2, K/2); } Complexity: T(N,K) = 2T(N/2, K/2) + cN Recursion tree has lg K levels, each requires cN time. So complexity is c N lg K.

6. 11/12/02CSE 202 - Shortest Paths CSE 202 - Algorithms Shortest Paths Problems

7. CSE 202 - Shortest Paths7 Shortest Paths Problems Given a weighted directed graph, is a path of weight 29 from u to z. is another path from u to z; it has weight 16 and is the shortest path from u to z. u z t y w v x 5 1 3 2 8 6 1 1 10 4 13 4 2 9

8. CSE 202 - Shortest Paths8 Variants of Shortest Paths Problems A. Single pair shortest path problem Given s and d, find shortest path from s to d. B. Single source shortest paths problem Given s, for each d find shortest path from s to d. C. All-pairs shortest paths problem For each ordered pair s,d, find shortest path. (1)and (2) seem to have same asymptotic complexity. (3) takes longer, but not as long as repeating (2) for each s.

9. CSE 202 - Shortest Paths9 More Shortest Paths Variants 1.All weights are non-negative. 2.Weights may be negative, but no negative cycles (A cycle is a path from a vertex to itself.) 3.Negative cycles allowed. Algorithm reports “-  ” if there is a negative cycle on path from source to destination (2) and (3) seem to be harder than (1). u w v 5 -6 3 2 uvw u058 v-303 w-60 u w v 5 6 -3 2 uvw u0-- -- v  -- -- w  -- --

10. CSE 202 - Shortest Paths10 Single Source Shortest Paths Non-negative weights (problem B-1): From source, construct tree of shortest paths. Why is this a tree? Is this a Minimum Spanning Tree? Basic approach: label nodes with shortest path found so far. Relaxation step: choose any edge (u,v). If it gives a shorter path to v, update v ’s label. 9 4 5 s 1 9 4 5 2 9 7 4 5 s 1 9 4 5 2 9 relaxing this edge improves this node

11. CSE 202 - Shortest Paths11 Single Source Shortest Paths Simplest strategy: –Keep cycling through all edges –When all edges are relaxed, you’re done. What is upper bound on work? Improvement: We mark nodes when we’re sure we have best path. Key insight: if you relax all edges out of marked nodes, the unmarked node that is closest to source can be marked. This gives Dijkstra’s algorithm Relax edge (u,v) only once - just after u is marked. Keep nodes on min-priority queue – need E Decrease-Key’s. Gives O(E lg V) (or O(E + V lg V) using Fibonacci heap.) There’s a simple O(V 2 ) implementation – when is it fastest?

12. CSE 202 - Shortest Paths12 Single Source Shortest Paths Negative weights allowed (problem B-2 and B-3) Why can’t we just add a constant to each weight? Why doesn’t Dijkstra’s algorithm work for B-2 ? Bellman-Ford: Cycle through edges V-1 times. O(VE) time. PERT chart: Arbitrary weights, graph is acyclic: Is there a better algorithm than Bellman-Ford?

13. CSE 202 - Shortest Paths13 All-pairs Shortest Paths Naïve #1: Solve V single-source problems. –O(V 2 E) for general problem. –O(V 3 ) or O(V E lg V) for non-negative weights. Naïve #2: Let d k (i,j) = shortest path from i to j involving  k edges. d 1 (i,j) = is original weight matrix. Compute d k+1 ’s from d k ’s by seeing if adding edge helps: d k+1 (i,j) = Min { d k (i,m) + d 1 (m,j) } Hmmm...this looks a lot like matrix multiplication If there are no negative cycles, d V-1 (i,j) is solution. If there is a negative cycle, then d V-1  d V Complexity is O(V 4 )

14. CSE 202 - Shortest Paths14 All-pairs Shortest Paths No negative cycles – Divide and Conquer says: “Shortest path from a to b with at most 2 k+1 hops is a shortest path from a to c with at most 2 k hops followed by one from c to b with at most 2 k hops.” T(k+1) = T(k) + V 3 so T(lg V) is O(V 3 lg V). This also looks like matrix multiplication, using d 2k = d k x d k instead of d k+1 = d k x d 1

15. CSE 202 - Shortest Paths15 All-pairs Shortest Paths No negative cycles – Dynamic Programming says: “Number cities 1 to V. Shortest path from a to b that only uses first cities 1 to k+1 as intermediate points is a path that goes from a to k+1 using only cities 1 to k, followed by a path going from k+1 to b (or shortest from a to b avoiding k+1 altogether).” T(k+1)= T(k) + cV 2, and T(0) is 0. Thus, T(V) is O(V 3 ). This is the Floyd-Warshall algorithm

16. CSE 202 - Shortest Paths16 Johnson’s All-Pairs Shortest Paths Motivation: for sparse non-negative weights, “O(V E lg V)” is better than “O(V 3 )” We’ll convert “arbitrary weights” (C3) to “non- negative” (C1) problem via reweighting. Dijkstra V times Floyd-Warshall 5 -3 8 -10 1 -3-3 8 0 4 -9 0 Starting bonus or finishing penalty

17. CSE 202 - Shortest Paths17 Johnson’s Algorithm Reweighting can make all edges non-negative Make node-weight(x) = shortest path to x from anywhere. Sounds like all-pairs problem Slick trick use Bellman-Ford (only O(VE) time) 5 -3 8 -10 0 5 10 2 3 0 0 0 4 5 -3 8 -10 s 0 0 0 0

18. CSE 202 - Shortest Paths18 Glossary (in case symbols are weird)            subset  element of  infinity  empty set  for all  there exists  intersection  union  big theta  big omega  summation  >=  <=  about equal  not equal  natural numbers(N)  reals(R)  rationals(Q)  integers(Z)