Trees, Trees and More Trees

Transcription

1 Trees, Trees and More Trees August 9, 01 Andrew B. Kahng

2 How You ll See Trees in CS Trees as mathematical objects Trees as data structures Trees as tools for complexity analysis Trees that are good in some way (and, algorithms to construct such trees) (many other contexts: game theory, bioinformatics, CAD, AI, machine learning, networking, )

3 1. Trees as Mathematical Objects Starting concept: graph Graph: G = (V,E) V = set of vertices E = set of edges graph with 7 vertices, 6 edges This graph is not connected, and it has a cycle. Connected: Between any pair of vertices, there is a path of edges. Cycle: A path of edges that begins and ends at the same vertex.

4 Tree = Special Kind of Graph A tree is a graph with n vertices that: (i) is connected (ii) has no cycles (iii) has n 1 edges tree with 7 vertices, 6 edges Fact: Any two of the above properties implies the third. E.g., If a graph with n vertices is connected and has no cycles, then it has n 1 edges. Can we prove this???

5 You want proof? I ll give you proof!

6 Claim: If a graph with n vertices connected and has no cycles, then it has n 1 edges. Proof Strategy:? Strong Induction (I.H.) Assume that the Claim is true for n = 1,, k 1. Consider a graph with n = k vertices that is connected and has no cycles. (k 1 1) + (k 1) + 1 = (k 1 + k ) 1 = k 1 k 1 k 1 vertices, connected, no cycles k 1 1 edges k k 1 vertices, connected, no cycles k 1 edges

7 Binary tree:. Trees as Data Structures Has a root vertex Every vertex has at most children Binary Search Tree Application: Search a set of keys for a given key value if key = root.key then success else if key < root.key then search left subtree else search right subtree Search for Search for 17 Search for 8

8 How Many Search Steps Are Needed? How many levels can a binary tree with n nodes have? (convention: root is at level 0) At most:? At least:? n 1 levels log n levels

9 . Trees as Tools for Complexity Analysis In the worst case, how hard can it be to sort n numbers using comparisons? Input: sequence of numbers, say, <A, B, C> Sorting == Identifying a Permutation Y B < C Y N A < B Y N A < C If #leaves a binary Tree of Comparisons tree has is at n! leaves, least = Sorting #permutations then the maximum Algorithm! (why?), which leaf is level? must be at least? N A < B < C Y A < C N B < A < C Y B < C N A < C < B C < A < B B < C < A C < B < A Leaf: A vertex with no children.

10 Binary Trees: #levels vs. #leaves Claim: If every leaf vertex in a binary tree has level at most H (= height ), then the tree has at most H leaves. (How can we prove this?) Proof Strategy =? Hint: = T T L T R What this will buy us: If a binary tree of comparisons has at least n! leaves, then its height must be at least log (n!) this gives us a lower bound on the worst-case number of comparisons needed to sort n elements! (you ll see this in CSE 0, 1, 101)

11 I think you should be more explicit here in step two.

12 4. Trees That Are GOOD For Something (A) Connecting cities with minimum road construction (and, doing this fairly) (and, adding junctions) (B) Synchronizing logic on a chip with minimum skew

13 Kinds of Algorithms Solution exact approximate Speed fast slow Short and sweet Slowly but surely Quick and dirty Too little, too late

14 The Minimum Spanning Tree Problem Given: graph of cities and distances B B A C A C E 4 D E 4 D Problem: Make all cities reachable from each other with minimum road construction.

15 The Shortest Paths Problem B B A C A C E 4 D E 4 D Problem: Compute the shortest routes from A to every other city.

16 The Shortest Paths Problem 4 E A D B C Problem: Compute the shortest routes from B to every other city. 4 E A D B C 5 0 What do these green numbers represent?

17 The Shortest Paths Problem 4 E A D B C Exploration parties set out from B, always moving at constant speed. When a party is the first to reach some city, they call home, then split up and continue exploring. The times of the phone calls are the shortest-path distances from B. 4 E A D B C 5 0 (You will soon learn this as Dijkstra s Algorithm!)

18 Minimum Spanning Tree, Again 4 4 E A D B C E A D B C Start from an arbitrary vertex, and add the shortest incident edge to the growing tree that does not complete a cycle. (If the MST is unique, then the starting vertex doesn t matter ) (You will soon learn this as Prim s Algorithm!)

19 Nearly the Same Algorithm! Prim: Iteratively add the edge e(u,v) to T, where u is in T and v is not yet in T, to minimize d u,v (d u,v = (u,v) distance) Dijkstra: Iteratively add the edge e(u,v) to T, where u is in T and v is not yet in T, to minimize d u,v + l u (l u is u s shortest path cost) Both algorithms build trees, in similar greedy ways! Prim s Algorithm constructs a Minimum Spanning Tree Dijkstra s Algorithm constructs a Shortest Path Tree If Prim s Algorithm and Dijkstra s Algorithm had a baby Iteratively add the edge e(u,v) to T, where u is in T and v is not yet in T, to minimize d u,v + c l u // 0 c 1 (food for thought!)

20 Algorithms Solution exact approximate Speed fast slow Short and sweet Slowly but surely Quick and dirty Too little, too late

21 The Non-Attacking Queens Problem Put N queens on an N x N chessboard such that no queen attacks another queen How many placements are possible? How can we go through them systematically? A queen in chess can attack any square on the same row, column, or diagonal. Given an N x N chessboard, we want to place N queens onto squares of the chessboard, such that no queen attacks another queen. This example: placement (red squares) of N = 4 mutually non-attacking queens.

22 Can Represent Solutions Using a Tree

23 Algorithms Solution exact approximate Speed fast slow Short and sweet Slowly but surely Quick and dirty Too little, too late

24 From Spanning Trees to Steiner Trees The red dot is called a Steiner point (= an intermediate junction ) The maximum cost savings from adding Steiner points = min ratio of (min Steiner tree cost) / (min spanning tree cost) depends on the metric, or distance function Cf. Manhattan, Euclidean, Chebyshev, Adapted from Prof. G. Robins, UVA

25 Minimum Steiner Tree is a Hard Problem (Minimum Spanning Tree was Easy) You ll learn about hard vs. easy in CSE101, CSE105 How would you approach finding a low-cost Steiner tree? Adapted from Prof. G. Robins, UVA

26 Iterated 1-Steiner Algorithm (1990) Given a pointset S, what point p minimizes MST(S {p}) Algorithmic idea: Iterate! (greedily) In practice: solution cost is within 0.5% of OPT on average Adapted from Prof. G. Robins, UVA

27 Light AND Shallow Trees (1991) What is the Minimum Spanning Tree? What is the Shortest Path Tree (with center point as source)? What tree is both light and shallow? 1991 Algorithm: Input: points, or graph > 0 Output: tree with radius (1 + ) * OPT cost (1 + / ) * OPT Adapted from Prof. G. Robins, UVA

28 Minimizing Skew of a Tree Goal: Deliver a signal (e.g., a clock pulse in a chip) to all leaves of the tree at exactly the same time Skew = difference in signal arrival time at the leaves of the tree leaves Critical problem in high-performance chip design How would you construct a low-skew tree???

29 Low-Skew Trees (1990) Algorithmic ideas: recursive geometric matching (= pairing up points) + finding balance points Adapted from Prof. G. Robins, UVA

30 Trees, Trees and More Trees Trees as mathematical objects Trees as data structures Trees as tools for complexity analysis Trees that are good in some way (and, algorithms to construct such trees) Good = Light, Shallow, Light and Shallow, Low-Skew, Spanning trees, Steiner trees Many contexts: game theory, bioinformatics, CAD, AI, machine learning, networking, feel free to ask!

31 Thank You!