CSE 373: Data Structures and Algorithms. Disjoint Sets. Autumn Shrirang (Shri) Mare

Transcription

1 CSE 373: Data Structures and Algorithms Disjoint Sets Autumn 2018 Shrirang (Shri) Mare Thanks to Kasey Champion, Ben Jones, Adam Blank, Michael Lee, Evan McCarty, Robbie Weber, Whitaker Brand, Zora Fung, Stuart Reges, Justin Hsia, Ruth Anderson, and many others for sample slides and materials... 1

2 Announcements Final exam info (logistics, topics covered, and practice material) is on course website. Homework 6 is due this Friday at noon. Homework 7 will be posted this Friday evening. - Fill out the team sign up form by tomorrow 5pm to get the repo in time. - Fill out the partner pool form by tomorrow 5pm to get assigned to a partner. CSE 373 AU 18 2

3 Today Trees and Forests Kruskal s algorithm Disjoint Set ADT CSE 373 AU 18 3

4 Trees and Forests A tree is an undirected, connected, and acyclic graph. A graph, however, can have unconnected vertices, or multiple trees. Collection of trees is a called a forest. A forest is any undirected and acyclic graph. - By definition a tree is a forest CSE 373 AU 18 4

5 Worksheet question 1 Review CSE 373 AU 18 5

6 Worksheet question 2a Review Step Components Edge Include? 6 A 3 1 B E 1 C D 5 F CSE 373 AU 18 6

7 <latexit sha1_base64="d8mbhvzwryhpx9trnkdkkhjog7i=">aaadm3icbvlbthrbej1hveb6a300jhu3jjgy3ovfh4leec8jbhzi2a2p6a7z6ddtpenuadfj/ph/4puv+hxwzo6cgjvmuqzup+pujyvwpns7p+of1q3bd+4u3mvff/dw0eol5sch3pzouf9ybd1rgp60mtqpkmg6khxhnmg6te43a//hgtmvrnkp44kgoy6mspxawkat5xhvknbimqr1ydovslyjyxfv2b5slubumdunv/pt1jnq22grwfakpdglgaiuuughvt8jceugjbpogwqlcqgcshlhdzkwj/fwbuctgesipcrjocc1yti/zv3vlkg9sfq+groxxl7jianwsximu5nwjacrcaboanyqkk3jmcylmr/od6w8kmj1uk+qexzz258hlivsnlnnaenctq7zmno/i7w3cidthm52+s4iag/iycv8nsx1umvdrucm0pspnwgmu7yg1wnprzkzlndofyuukkgjzyk9fshocuthrbpuyw+rzvetwgglhnq6/hoodn3bzuafuffjpohihepmr/tq4/98x2vi3w4rzyoykbftoltudyd8rcy7ixh0mbuuplysidj0kalf2lwuos+t6+bcu8ldw1lpyf+8rjpnxnpexoapkrtoimqyav16yolygrsx7tzt3lto6e3lyh2tx/rx9c7guxnzi9gz6ew0gvwin9fg9chajfqrih/ev+lf8z/w89zm62pr8zr0iz7lpi2uskv/f4tukjq=</latexit> <latexit sha1_base64="d8mbhvzwryhpx9trnkdkkhjog7i=">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</latexit> <latexit sha1_base64="d8mbhvzwryhpx9trnkdkkhjog7i=">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</latexit> Worksheet question 2b Review 1: function Kruskal(Graph G) 2: initialize each vertex to be a component 3: sort all edges by weight 4: for each edge (u, v) in sorted order do 5: if u and v are in di erent components then 6: add edge (u,v) to the MST 7: update u and v to be in the same component 8: end if 9: end for 10: end function CSE 373 AU 18 7

8 <latexit sha1_base64="d8mbhvzwryhpx9trnkdkkhjog7i=">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</latexit> <latexit sha1_base64="d8mbhvzwryhpx9trnkdkkhjog7i=">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</latexit> <latexit sha1_base64="d8mbhvzwryhpx9trnkdkkhjog7i=">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</latexit> Kruskal s Algorithm 1: function Kruskal(Graph G) 2: initialize each vertex to be a component 3: sort all edges by weight 4: for each edge (u, v) in sorted order do 5: if u and v are in di erent components then 6: add edge (u,v) to the MST 7: update u and v to be in the same component 8: end if 9: end for 10: end function CSE 373 AU 18 8

9 Minimum spanning forest Given a forest, find a set of minimum spanning trees for each tree in the given forest. Question: Which algorithm (Prims or Kruskals) would you use to find a minimum spanning forest? CSE 373 AU 18 9

10 Disjoint Set ADT CSE 373 AU 18 10

11 New ADT state Count of Elements behavior Set ADT Set of elements - Elements must be unique! - No required order create(x) - creates a new set with a single member, x add(x) - adds x into set if it is unique, otherwise add is ignored remove(x) removes x from set size() returns current number of elements in set state Disjoint-Set ADT Set of Sets - Disjoint: Elements must be unique across sets - No required order - Each set has representative Count of Sets behavior makeset(x) creates a new set within the disjoint set where the only member is x. Picks representative for set findset(x) looks up the set containing element x, returns representative of that set union(x, y) looks up set containing x and set containing y, combines two sets into one. Picks new representative for resulting set D B C A D C F B A G H CSE 373 AU 18 11

12 Example new() makeset(a) makeset(b) makeset(c) makeset(d) makeset(e) findset(a) findset(d) union(a, c) Rep: 0 a Rep: 4 e Rep: 2 c Rep: 1 b Rep: 3 d CSE 373 AU 18 12

13 Example new() makeset(a) makeset(b) makeset(c) makeset(d) makeset(e) findset(a) findset(d) union(a, c) union(b, d) Rep: 0 a c Rep: 4 e Rep: 1 b Rep: 3 d CSE 373 AU 18 13

14 Example new() makeset(a) makeset(b) makeset(c) makeset(d) makeset(e) findset(a) findset(d) union(a, c) union(b, d) findset(a) == findset(c) Rep: 0 a c Rep: 4 e Rep: 1 b d findset(a) == findset(d) CSE 373 AU 18 14

15 Implementation Disjoint-Set ADT state Set of Sets - Disjoint: Elements must be unique across sets - No required order - Each set has representative Count of Sets behavior makeset(x) creates a new set within the disjoint set where the only member is x. Picks representative for set findset(x) looks up the set containing element x, returns representative of that set union(x, y) looks up set containing x and set containing y, combines two sets into one. Picks new representative for resulting set state behavior TreeDisjointSet<E> Collection<TreeSet> forest Dictionary<NodeValues, NodeLocations> nodeinventory makeset(x)-create a new tree of size 1 and add to our forest findset(x)-locates node with x and moves up tree to find root union(x, y)-append tree with y as a child of tree with x state behavior TreeSet(x) TreeSet<E> SetNode overallroot add(x) remove(x, y) getrep()-returns data of overallroot state E data SetNode(x) SetNode<E> Collection<SetNode> children behavior addchild(x) removechild(x, y) CSE 373 AU 18 15

16 Implement makeset(x) makeset(0) makeset(1) makeset(2) makeset(3) forest state behavior TreeDisjointSet<E> Collection<TreeSet> forest Dictionary<NodeValues, NodeLocations> nodeinventory makeset(x)-create a new tree of size 1 and add to our forest findset(x)-locates node with x and moves up tree to find root union(x, y)-append tree with y as a child of tree with x makeset(4) makeset(5) Worst case runtime? O(1) CSE 373 AU 18 16

17 Implement union(x, y) forest union(3, 5) state TreeDisjointSet<E> Collection<TreeSet> forest Dictionary<NodeValues, NodeLocations> nodeinventory behavior makeset(x)-create a new tree of size 1 and add to our forest findset(x)-locates node with x and moves up tree to find root union(x, y)-append tree with y as a child of tree with x > -> -> -> -> -> CSE 373 AU 18 17

18 Implement union(x, y) union(3, 5) union(2, 1) forest state TreeDisjointSet<E> Collection<TreeSet> forest Dictionary<NodeValues, NodeLocations> nodeinventory behavior makeset(x)-create a new tree of size 1 and add to our forest findset(x)-locates node with x and moves up tree to find root union(x, y)-append tree with y as a child of tree with x > -> -> -> -> -> CSE 373 AU 18 18

19 Implement union(x, y) union(3, 5) union(2, 1) union(2, 5) forest state TreeDisjointSet<E> Collection<TreeSet> forest Dictionary<NodeValues, NodeLocations> nodeinventory behavior makeset(x)-create a new tree of size 1 and add to our forest findset(x)-locates node with x and moves up tree to find root union(x, y)-append tree with y as a child of tree with x > -> -> -> -> -> CSE 373 AU 18 19

20 Implement union(x, y) union(3, 5) union(2, 1) union(2, 5) forest state behavior TreeDisjointSet<E> Collection<TreeSet> forest Dictionary<NodeValues, NodeLocations> nodeinventory makeset(x)-create a new tree of size 1 and add to our forest findset(x)-locates node with x and moves up tree to find root union(x, y)-append tree with y as a child of tree with x CSE 373 AU 18 20

21 TreeDisjointSet<E> Implement findset(x) findset(0) findset(3) findset(5) forest state Collection<TreeSet> forest Dictionary<NodeValues, NodeLocations> nodeinventory behavior makeset(x)-create a new tree of size 1 and add to our forest findset(x)-locates node with x and moves up tree to find root union(x, y)-append tree with y as a child of tree with x Worst case runtime? O(n) Worst case runtime of union? O(n) CSE 373 AU 18 21

22 Improving union Problem: Trees can be unbalanced Solution: Union-by-rank! - let rank(x) be a number representing the upper bound of the height of x so rank(x) >= height(x) - Keep track of rank of all trees - When unioning make the tree with larger rank the root - If it s a tie, pick one randomly and increase rank by one rank = 0 rank = 2 rank = 0 rank = 0 rank = CSE 373 AU 18 22

23 Worksheet question 3 Given the following disjoint-set what would be the result of the following calls on union if we add the union-by-rank optimization. Draw the forest at each stage with corresponding ranks for each tree. rank = 2 rank = 0 rank = 2 rank = union(2, 13) union(4, 12) union(2, 8) CSE 373 AU 18 23

24 Worksheet question 3 Given the following disjoint-set what would be the result of the following calls on union if we add the union-by-rank optimization. Draw the forest at each stage with corresponding ranks for each tree. rank = union(2, 13) union(4, 12) union(2, 8) Does this improve the worst case runtimes? 1 5 findset is more likely to be O(log(n)) than O(n) CSE 373 AU 18 24

25 Improving findset() Problem: Every time we call findset() you must traverse all the levels of the tree to find representative Solution: Path Compression - Collapse tree into fewer levels by updating parent pointer of each node you visit - Whenever you call findset() update each node you touch s parent pointer to point directly to overallroot rank = 3 rank = 2 findset(5) 8 8 findset(4) Does this improve the worst case runtimes? findset is more likely to be O(1) than O(log(n)) CSE 373 AU 18 25