Final Exam. Jonathan Turner 5/12/2010. CS 542 Advanced Data Structures and Algorithms
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1 CS 542 Advanced Data Structures and Algorithms Final Exam Jonathan Turner 5/12/ (10 points) In the analysis that establishes an O(log log n) bound on the amortized time per operation for the partition data structure, a node x is defined to be dominant if Δ(x)>2Δ(y) for all ancestors y of x. Define Δ(x). 30 a 29 b The diagram at right shows one tree in a partition data structure; the numbers represent ranks. Give a numerical lower bound on the number of nodes in the tree d c 20 e List the dominant nodes in the tree. 16 f 3 g Suppose a find operation is done at node g. For each non-root node u on the find path that is not dominant before the find, give the ratio of the new value of Δ(u) to the old value
2 (10 points) In the Fibonacci heap shown below, the numbers are the key values. How many credits are needed to ensure that the credit invariant used in the amortized analysis is satisfied? Show the heap that results from performing a deletemin on the heap. How many credits are needed to maintain the invariant following the deletemin? a 3 b 2 c 4 d e 6 f 7 g h i 9 j k l o p m n q 1
3 3. (10 points) The diagram below shows a dual-key search tree that uses a differential representation for the key2 values. 7 a 1,2 key 1 Δkey 2,Δmin 2 3 e 3,1 9 b 0,0 1 h 0,2 6 f 0,0 13 c 1, g 0,1 11 j 0,3 d What value is returned by findmin(3,8,a)? 0,0 Show how the structure of the tree is changed by the operation access(4,a), assuming a selfadjusting search tree is used as the basis for the dual-key search tree. You need not show updated key values in this case, but be sure to label each node. Suppose we performed 100 operations on a dual-key search tree with 1000 nodes. Give a numerical upper bound on the number of splay steps that would be performed in this case
4 4. (10 points) The picture below shows an implementation of a dynamic tree, using linked paths represented by binary search trees. Draw the actual tree that this represents (with vertex costs). Assuming the actual tree is being used in a maximum flow computation, what is the residual capacity of the path from s to t in the actual tree. Which edges become saturated if the maximum possible amount of flow is added to this path? a 2,3 Δmin, Δcost e 0,1 t 1,97 k 3,3 b 0,0 d 0,0 c 9,0 s 1,0 f 0,0 m 3,0 h 2,0 g 4,0 n 2,0 i 3,0-4 -
5 5. (15 points) In the analysis of the general preflow-push method for max flows, a potential function is used to account for the number of steps in which flow is added to an edge, but the edge is not saturated. The potential function is defined to be the sum of the distance labels of the unbalanced vertices (that is, the vertices that have positive excess flow). Suppose we have a graph with 100 vertices and 1000 edges. Give a numerical upper bound on the value of the potential function and explain your answer. Explain why every step that adds flow to an edge (u,v) without saturating it, causes the potential function to decrease. How is the potential function affected by a step that adds flow to edge (u,v) and saturates it? Give a big-o bound on the sum of the increases in the potential that could result from all such steps, assuming a graph with n vertices and m edges How is the potential function affected by a step that changes the label of a vertex u. Give a big-o bound on the sum of the increases in the potential that could result from all such steps
6 6. (15 points) The diagram below shows an instance of the min-cost, max-flow problem with a preflow and a set of vertex labels. cost,capacity,flow 0 0 excess a 2,3,0 c 2 5 5,5,2 3,2,0 2 1,2,0-2 s 3,2,2 4,2,2 t -3 3,4,2 5,4, ,4,2 b 1,2,0 d label 0 1 Show that the given set of labels is not valid. Explain how you can make the labels valid by adding flow to selected edges. List all the edges for which the flow must be changed and give the new flow value for each edge. After making these changes, does the resulting residual graph have a negative cost cycle? If so, list the vertices in the cycle. If not, explain why it does not
7 7. (15 points) Suppose we wanted to add the following operation to the dynamic trees data structure. ancestor(x,y) returns true if y is an ancestor of x, else false Describe how you could implement this operation using the path set operations and the expose operation. In the breadth-first scanning algorithm for finding shortest paths, cycles are detected using a pass-counting method. Suppose that we want to detect a negative cycle in the parent pointers at the time the cycle is about to be created? This can be done by checking to see if a vertex x is an ancestor of y in the shortest path tree, whenever we re about to make x the parent of y. The main loop of the basic version of the breadth-first-scanning algorithm is shown below. Show how you would modify it to use a dynamic tree data structure in place of the parent pointers, and use the ancestor operation to detect a negative cycle. do queue [ ] od; v := queue(1); queue := queue[2..]; for [v,w] out(v) rof; if dist(v)length(v,w)<dist(w) fi; p(w) := v; dist(w) := dist(v)length(v,w); if w queue queue := queue & [w]; fi; What is the asymptotic running time for this version of the algorithm? - 7 -
8 8. (15 points) Define a k-matching of a bipartite graph to be a subset of its edges that defines a subgraph with maximum degree k (so, we can view an ordinary matching as a k-matching for k=1). Describe ( in words) an algorithm that can find a maximum size k-matching of any connected, bipartite graph, for any integer k 1 and explain briefly why it works. What is the running time of your algorithm? - 8 -
9 9. (10 points) The diagram below shows an intermediate state in the execution of the Edmonds-Karp algorithm for finding a matching in a graph. a g m st s b h ts n t c i o u d j p v jd e k x w In this state, what is the value of origin(find(e))? What is origin(find(m))? If the edge ot is chosen next, the algorithm finds an augmenting path. What is that path? Suppose instead, that edge xw is chosen next, causing a new blossom to be formed. Outline the vertices that are included in the new blossom and for each odd vertex z in the blossom, write the value of bridge(z) on the diagram. If the edge ni is chosen after the blossom is formed, the algorithm finds an augmenting path. What is that path? - 9 -
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