prerequisites: 6.046, 6.041/2, ability to do proofs Randomized algorithms: make random choices during run. Main benefits:

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1 Itro Admiistrivia. Sigup sheet. prerequisites: 6.046, 6.041/2, ability to do proofs homework weekly (first ext week) collaboratio idepedet homeworks gradig requiremet term project books. questio: scribig? Radomized algorithms: make radom choices durig ru. Mai beefits: speed: may be faster tha ay determiistic eve if ot faster, ofte simpler (quicksort) sometimes, radomized is best sometime, radomized idea leads to determiistic algorithm Distiguish average-cast aalysis Probabilistic aalysis assumig radom iput radomized algorithms do ot assume radom iputs so aalyses are more applicable We do t really use radom umbers. But radomized algorithms break patters we do t kow are there. determiistic algorithm: works well except a few specific cases. 1

2 But those are the oes you will ecouter (Murphy)! radomized: almost always works well o ay case but sometimes does bad o ay case, so risky for life-threateig errors. Course objective: Radomizatio is a geeral techique. Applies to all areas of CS. Uderlyig it is a commo set of tools. Goal is to give familiarity with those tools so you ca apply them to your ow problems. To preset tools, we draw appliatios from may areas of CS: data structures, geometric algos, graph algos, parallel ad distributed, umber theory. Because so may, oly a brief taste of each. But sufficiet to go o aloe. Basic methodologies. Avoidig adversarial iputs sorted quicksort list a kid of radom reorderig (geometry BSP) hashig to same buckets olie algorithms ote: adversarial may mea well structured i.e. atural figerpritig/verificatio geerate short radom figerprits for thigs faster tha comparig thigs almost every figerprit works so a radom oe works 2

3 radom samplig. graph algs, computatioal geometry, media fast way to fid typical members solve represetative subproblem fast extrapolate to solutio of origial problem load balacig radomizatio spreads thigs out uiformly parallel algs, routig, hashig symmetry breakig radom decisios keep everyoe from doig the same thig etheret deadlocks avoidace i distributed systems (MUST radomize) Probabilistic existece proofs thought experimet prove a object is build with positive probability guaratees object exists makes search for algo worthwhile. Today: 2 really basic priciples: liearity of expectatio product of evet probabilities (idepedece) The some fudametal ideas: Kids of radomized algorithms a bit of complexity 3

4 Quicksort Items S 1,..., S to be sorted suppose could pick middle elemet: T () = 2T (/2) + O() = O( log ) works sice divides ito much smaller subproblems pickig middle is hard. But a almost middle elemet is OK. pick radom elemet. probably ear middle ad divides problem i two boud expected umber of comparisos C X ij = 1 if compare i to j liearity of expectatio: E[C] = E[X ij ] E[X ij ] = p ij Cosider smallest recursive call ivolvig both i ad j. pivot must be oe of S i,..., S j. all equally likely S i ad S j get compared if pivot is S i or S j probability is at most 2/(j i + 1) (may have outer elemets) aalysis: p ij 2/(j i + 1) i=1 j>i i=1 j>i i+1 = 2/k i=1 k=1 2 1/k i=1 k=1 2H 4

5 BSP (Defie H, claim O(log ).) = O( log ). aalysis holds for every iput, does t assume radom iput we proved expected. ca show high probability how did we pick a radom elemets? Depeds o model. algorithm always works, but might be slow. liearity of expectatio. hat check problem Rederig a image reder a collectio of polygos (lies) paiters algorithm: draw from back to frot; let frot overwrite eed to figure out order with respect to user defie BSP. BSP is a data structure that makes order determiatio easy Build i preprocess step, the reder fast. Choose ay hyperplae (root of tree), split lies oto correct side of hyperplae, recurse If user is o side 1 of hyperplae, the othig o side 2 blocks side 1, so pait it first. Recurse. time=bsp size sometimes must split to build BSP how limit splits? autopartitios radom auto 5

6 aalysis idex (u, v) = k if k lies block v from u u v if v cut by u auto probability 1/(1 + idex (u, v)). tree size is (by liearity of E) + 1/idex (u, v) 2H result: exists size O( log ) auto gives radomized costructio equally importat, gives probabilistic existece proof of a small BSP so might hope to fid determiistically. MiCut the problem cotractio coditioally idepedet evets give/aalyze repetitio for better success probability (idepedet evets) faster implemetatio later Mote Carlo vs. Las Vegas tur LV to MC by trucatig tur MC to LV by certifyig. if ca t certify, dagerous! u 6