Lecture 4: Advanced Data Structures

Transcription

1 Lecture 4: Advanced Data Structures Prakash Gautam

2 Agenda Heaps Binomial Heap Fibonacci Heap Hash Tables Bloom Filters Amortized Analysis 2

3 Binomial Tree A binomial tree of order k is defined recursively: Order 0: Single Node Order k: One binomial tree of order k-1 linked to another of order k-1 3

4 Binomial Tree: Properties Given an order k binomial tree Bk, Its height is k It has 2k nodes The degree of its root is k Deleting its root yields k binomial trees: Bk-1, Bk-2,.. B1, B0 4

5 1. Binomial Heap A binomial heap is a sequence of binomial trees such that: Each tree is heap-ordered There is either 0 or 1 binomial tree of order k Collection of ordered binomial trees Required for efficient union of n heaps 5

6 Binomial Heap: Properties Given a binomial heap with n nodes: The node containing the min element is a root of B0, B1,, or Bk It contains the binomial tree Bi, iff bi=1, where bk.b2b1b0 is binary representation of n It has <= Floor[logn] + 1 binomial trees Its height <= Floor [logn] 6

7 Binomial Heap: Meld Meld Operation: Given two binomial heaps H1 & H2, replace with a binomial heap H that is the union of two If H1 & H2 are both binomial trees of order k: Connect roots of H1 & H2 Choose node with smaller key to be root of H 7

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10 Binomial Heap: Meld Solution: Analogous to Binary Addition Running Time: O(log n) = O(# Trees) 10

11 Binomial Heap: Extract the Minimum Delete the node with minimum key in binomial heap, H Find root x with min key in root list of H & Delete H = Broken Binomial Tree H = Meld(H, H) Running Time: O(log n) 11

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13 Binomial Heap: Decrease Key Given a handle to an element x in H, decrease its key to k Suppose x is in binomial tree, Bk Repeatedly exchange x with its parent until heap order is restored Running Time: O(log n) 13

14 Binomial Heap: Insert Given a binomial heap H, Insert an element x H = MakeHeap( ) H = Insert (H, x) H = Meld(H, H) Running Time: O(log n) 14

15 Fibonacci Numbers & NATURE 15

16 2. Fibonacci Heap Forest of min-heaps Lazy binomial heaps Tidying up; do it on-the-fly when search for the MIN (Fibonacci Heap) Immediately tidy up after INSERT or MERGE (Binomial Heap) Collection of unordered binomial trees Required for efficient union as well as efficient operations Maintains pointer to MIN element Nodes can be marked but roots are always unmarked 16

17 Heap ordered trees 17

18 Maintain pointer to MIN element: O(1) 18

19 Set of marked node: Used to keep heaps flat 19

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21 Potential Function 21

22 Fibonacci Heap: INSERT Create a new Singleton tree Add to root list; update MIN pointer if necessary 22

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24 Fibonacci Heap: DELETE MIN Delete MIN; Meld its children into root list; update MIN Consolidate trees so that no two roots have same rank 24

25 Fibonacci Heap: DELETE MIN 25

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28 Fibonacci Heap: DECREASE KEY If heap order is not violated: Just decrease the key of x Otherwise, cut tree rooted at x and meld into root list And keep the tree as flat as possible 28

29 Cut the tree rooted at x, meld into root list & Unmark 29

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39 Linear Search vs Binary Search 39

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42 3. Hash Tables Hash Table Mostly it is like a modified array to store dataset. It is the special type of Data Structure Hash Function Function that can be used to map dataset of any size to dataset of fixed size which falls into the hash table The values returned by hash function are called hash values, hash codes, hash sums, or simply hashes Hashing Large keys are converted into small ones by using hash functions & the values are stored in DS called hash tables 42

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44 Basically, all of the operations on an element of a hash table should be O(1) However, this efficiency is only realized if each element maps to a unique position in the table A collision occurs when two or more elements map to the same location A hashing function that maps each element to a unique location is called perfect hashing function Perfect Hashing Function: Operations, O(1) Another Issue: How large the table should be? If we are assured of a dataset of size n and a perfect hashing function, then the table should be of size n A good rule of thumb is to make the table 150% the size of the dataset Another Issue: If dataset is dynamic? 44

45 Dynamic Resizing Dynamic resizing of a hash table involves creating a new, larger, hash table and then inserting all of the elements of the old table into the new one Deciding when to resize is the key to dynamic resizing One possibility is to resize when the table is full It is the nature of hash tables that their performance seriously degrades as they become full A better approach is to use a load factor The load factor of a hash table is the percentage occupancy of the table at which the table will be resized 45

46 Hashing Functions Direct Hashing No algorithmic manipulation No collision but not not suitable for large range of keys Modulo Division Hashing (Division Remainder Hashing) Works with any size of dataset If list size is Prime number: Less # of collisions h(key) = Key % Size Mid Square Hashing Folding Hashing Pseudo Random Hashing Y = ax + c, h(x) = Y % Size Subtraction Hashing h(key) = Key - c 46

47 Resolving Collisions Chaining Open Addressing Linear Probing : New Position = (Hash Value + 1) % Size Quadratic Probing: New Position = (Hash Value + i2) % Size Double Hashing h2(x)=r-(x % R), Where R is prime number smaller than hash table size New Position= (Hash Value + i * h2(x) ) % Size 47

48 3. Bloom Filters Distributed Data Google, Facebook, Twitter deal with large volume of data Hundreds of PetaBytes There is absolutely no way to store this on one computer Instead, data must be stored on multiple computers networked together Looking up Data Suppose, You are at Google implementing search When a search query is initiated, You have to be able to know which computer knows what pages to display for that query Network Latency: Suppose 2ms If you have one thousand computers to search, you can t just query each one & ask! 48

49 Bloom Filter: Similar to Hash Table x: An element S: Set of elements Input: (x, S) Output: T: if x in S F: if x not in S A Bloom Filter consists of vector of n boolean values, initially all set false, as well as k independent hash functions, h0, h1,, hk-1, each with range 0 to n-1 [F F F F F F F F F F] Initial Setup (n=10) 49

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51 Installing two elements For each element s in S, the boolean values with positions h0(s), h1(s),., hk-1(s), are set TRUE. Note: A particular boolean value may be set to true several times [F T F T F F T F F F] Initial Setup (n=10) 51

52 Error Types False Negative Answering not there on an element that is in the set (Never Happens for BF) False Positive Answering is there on an element that is not in the set We need to design the filter so that the probability of false positive is too small Stores a set of values in a way that may lead to false positives If BF says that an object is not present, It is definitely not present If BF says that an object is present, It may actually not be present 52

53 Bloom Filters & Networks It can be used to mitigate the networking problems To determine which computer has data: Look up that value in each Bloom Filter Call up just the computers that might have it Bloom Filter lookup is upto times faster than Network Query. So, It is used extensively in practice Data Structures make it possible to solve important problems at scale. You get to decide which problems we ll be using them for! 53

54 THANK YOU 54