The Hiring Problem. Informationsteknologi. Institutionen för informationsteknologi

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1 The Hrng Problem An agency gves you a lst of n persons You ntervew them one-by-one After each ntervew, you must mmedately decde f ths canddate should be hred You can change your mnd f a better one comes up later, but that wll cost

2 The Hrng Problem: straghtforward algorthm Always change your mnd f a better one shows up Gven n canddates, how many tmes wll we change our mnd? Worst case: n tmes Expected case:? Hre Hre Hre Hre Hre Insttutonen för nformatonsteknolog n

3 Worst case: n tmes (better and better canddates) 1 2 n

4 Expected case:? We need probablstc analyss Hre Hre Hre Hre Hre 1 2 n

5 Probablstc analyss Use an ndcator varable 1f canddate s hred X 0 otherwse E( X ) denotes the expected value of Expected number of re - hrng : E n 1 X X n 1 E( X )

6 Probablstc analyss Use an ndcator varable 1f canddate s hred X 0 otherwse E( X ) denotes the expected value of E( X ) Pr(canddate s hred) Pr(canddate s hred) Pr(canddate s better than all n Expected number of re - hrng : E 1 E( X ) n 1 1 ln n O(1) n 1 X X n 1 E( X ) before) 1

7 Probablstc analyss: problem of trust We assume that the agency s lst s randomly ordered What f not? (If we pay the agency each tme we re-hre...)

8 Soluton: Use a randomzed algorthm Shuffle the lst randmly before the ntervews Now, the probablstc analyss holds ndependent of the agency s behavour

9 The Hrng Problem: randomzed algorthm 1. Permute the lst randomly 2. Always change your mnd f a better one shows up Gven n canddates, how many tmes wll we change our mnd? (Worst case: n tmes) Who cares? Expected case: ln n + O(1) tmes

10 To thnk about 1. How do we permute an array randomly? 2. On-lne Hrng Problem: We can not change our mnd, we can only hre one. How do we do ths?

11 Balls and Bns Balls are thrown n bns, each ball s thrown randomly and ndependently n balls, b bns

12 Balls and Bns Balls are thrown n bns, each ball s thrown randomly and ndependently n balls, b bns Expected number of balls n a gven bn: n/b Expected number of balls untl a gven bn gets one: b Expected number of balls untl each bn has a ball: b ln b + O(1) If b=n (.e., we throw n balls) Expected maxmum number of balls n a bn : (ln n) Expected largest number of Expected value of all bns conseccutve empty bns : (ln n) (number of balls) 2 ( n)

13 General Idea of Hashng Store elements (key, value) n an array Use the hash functon to determne where each key s stored If the hash functon s good, the keys are ncely spread If two keys have the same hash functon, we have a collson, whch must be handled

14 Handlng collsons Channg Open adressng Double hashng: Start at h 1 Jump h 2

15 How do we select a good hash functon?

16 Unversal Hash functon A randomly selected hash fucton Works well wth hgh probablty for any set of keys Good example of randomzed algorthm

17 Unversal Hash functon Gven a set H of hash functons that maps keys nto 0...m. If for each par (x,y) the number of hash functons for whch h(x) = h(y) s at most H / m, then H s unversal

18 Theorem 11.3 If we store n keys nto a table of sze m usng channg, the expected length of the chan contanng key k s n / m (=α) Proof sketch: for each other key, the probablty of collson wth k s 1 / m

19 Fndng unversal hash functons s easy! Class H p,m conssts of all hash functons h(k) = ((ak+b) mod p) mod m where m s table sze, p s a prme, p > m a and b are random numbers Theorem 11.5: H p,m s unversal

20 Wth unversal hashng, expected cost per operaton s low. But what f we want the max cost per operaton to be low? (Let s say we wsh to construct a statc hash table to be stored on a CD-ROM and we want each search to be fast)