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Dynamic Sets and Data Structures Over the course of an algorithm’s execution, an algorithm may maintain a dynamic set of objects The algorithm will perform operations on this set –Queries –Modifying operations We must choose a data structure to implement the dynamic set efficiently The “correct” data structure to choose is based on –Which operations need to be supported –How frequently each operation will be executed

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Some Example Operations Notation –S is the data structure –k is the key of the item –x is a pointer to the item Search(S,k): returns pointer to item Insert(S,x) Delete(S,x): note we are given a pointer to item Minimum or Maximum(S): returns pointer Decrease-key(S,x) Successor or Predecessor (S,x): returns pointer Merge(S 1,S 2 )

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Basic Data Structures/Containers Unsorted Arrays Sorted Array Unsorted linked list Sorted linked list Stack Queue Heap

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Puzzles How can I implement a queue with two stacks? –Running time of enqueue? –Dequeue? How can I implement two stacks in one array A[1..n] so that neither stack overflows unless the total number of elements in both stacks exceeds n?

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Unsorted Array Sorted Array Unsorted LL Sorted LL Heap Search Insert Delete Max/Min Pred/Succ Merge

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Case Study: Dictionary Search(S,k) Insert(S,x) Delete(S,x) Is any one of the data structures listed so far always the best for implementing a dictionary? Under what conditions, if any, would each be best? What other standard data structure is often used for a dictionary?

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Case Study: Priority Queue Insert(S,x) Max(S) Delete-max(S) Decrease-key(S,x) Which data structure seen so far is typically best for implementing a priority queue and why?

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Case Study: Minimum Spanning Trees Input –Weighted, connected undirected graph G=(V,E) Weight (length) function w on each edge e in E Task –Compute a spanning tree of G of minimum total weight Spanning tree –If there are n nodes in G, a spanning tree consists of n-1 edges such that no cycles are formed

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Prim’s algorithm A greedy approach to edge selection –Initialize connected component N to be any node v –Select the minimum weight edge connecting N to V-N –Update N and repeat Dynamic set in Prim’s algorithm –An item is a node in V-N –The value of a node is its minimum distance to any node in N –A minimum weight edge connecting N to V-N corresponds to the node with minimum value in V-N (Extract minimum) –When v is added to N, we need to update the value of the neighbors of v in V-N if they are closer to v than other nodes in N (Decrease key)

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Maintain dynamic set of nodes in V-N If we started with node D, N is now {C,D} Dynamic set values of other nodes: –A, E, F: infinity –B: 4 –G: 6 Extract-min: Node B is added next to N Illustration ABC D EFG 1 2 2 3 4 5 5 6 10

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Node B is added to N; edge (B,C) is added to T Need to update dynamic set values of A, E, F –Decrease-key operation Dynamic set values of other nodes: –A: 1 –E: 2 –F: 5 –G: 6 Extract-min: Node A is added next to N Updating Dynamic Set ABC D EFG 1 2 2 3 4 5 5 6 10

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Node A is added to N; edge (A,B) is added to T Need to update dynamic set values of E –Decrease-key operation Dynamic set values of other nodes: –E: 2 (unchanged because 2 is smaller than 3) –F: 5 –G: 6 Updating Dynamic Set Again ABC D EFG 1 2 2 3 4 5 5 6 10

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Dynamic Set Analysis How many objects in initial dynamic set representation of V-N? How many extract-min operations need to happen? How many decrease-key operations may occur? Given all of the above, choose a data structure and tell me the implementation cost. –Time to build initial dynamic set –Time to implement all extract-min operations –Time to implement all decrease-key operations

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Kruskal’s Algorithm A greedy approach to edge selection –Initialize tree T to have no edges –Iterate through the edges starting with the minimum weight one Add the edge (u,v) to tree T if this does not create a cycle

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6 ABC D EFG 1 3 8 2 4 7 5 9 Example (A,B) (A,E) (B,E): cycle (B,C) (F,G) (C,G) (B,F): cycle (C,D) (D,G): cycle ABC D EFG 1 3 8 2 4 7 5 6 9 ABC D EFG 1 3 8 2 4 7 5 6 9 ABC D EFG 1 3 8 2 4 7 5 6 9

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Disjoint Set Data Structure Given a universe U of objects (nodes V) –Maintain a collection of disjoint sets S i that partition U –Find-set(x): Returns set S i that contains x –Merge(S i, S j ): Returns new set S k = S i union S j Disjoint Sets and Kruskal’s algorithm –Universe U is the set of vertices V –The sets are the current connected components –When an edge (u,v) is considered, we check for a cycle by determining if u and v belong to the same set 2 calls to Find-set(x) –If we add (u,v) to T, we need to merge the 2 sets represented by u and v. Merge(S u,S v )

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Analysis How do we initialize the universe? How many calls to find-set do we perform? How many calls to merge-set do we perform?

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Better data structures We need mergeable data structures that still support fast searches –Binomial heaps (ch. 19) –Fibonacci heaps (ch. 20) –Disjoint set data structures (ch. 21) linked lists forests

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Disjoint-set forests Representation –Each set is represented as a tree, nodes point to parent –Root element is the representative for the set, points to self or has null parent pointer –Height: maintain height of tree as an integer Operations –Makeset: make a tree with one node –Find: progress from current element to root element following links –Union: connect root of lower height tree to point to root of larger height tree Figures copied from Jeff Erickson, UIUC

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Naïve implementation Figure copied from Jeff Erickson’s slides at UIUC.

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union-by-rank or union-by-depth Figure copied from Jeff Erickson’s slides at UIUC. Leads to height of any tree of n nodes being at most O(lg n).

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Path Compression Figure copied from Jeff Erickson’s slides at UIUC. Leads to amortized cost of α(n), the inverse ackerman function. For all practical purposes, α(n) ≤ 4.

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Binomial Heaps Binomial Tree Binomial Heap –Figures copied from Dan Gildea, University of Rochester

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Key idea: Union in O(lg n) time

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Binomial Trees Tree B k has 2 k nodes. B k has height k. Children of the root of B k are B k-1, B k-2, …, B 0 from left to right. Max degree of an n-node binomial tree is lg n.

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Binomial Heap A binomial heap of n-elements is a collection of binomial trees with the following properties: –Each binomial tree is heap-ordered (parent is less than all children) –No two binomial trees in the collection have the same size –Number of trees will be O(lg n)

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Example Binomial Heap Binomial heap of 29 elements 29 = 11101 in binary.

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Minimum Operation Where does the minimum have to be? How can we find minimum in general? Running time?

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Union of 2 Binomial Heaps

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