 
Summary: Online and Approximation Algorithms Fall Semester, 2009/10
Final Exam: January 25, 2010
Lecturer: Prof. Yossi Azar
Write short but full and accurate answers. Each question should start on a new page and each of its parts
should not exceed a page. Solving 5 questions results in score 90. Two pages A4 (both sides each) are allowed.
1. Consider the ski rental problem (1 to rent, M to buy) with the following frequent skier program. After
(1 + )M times (for some fixed real number > 0, for example, = 0.7 or = 2.3 ) you have rented
the skis you get them for free.
(a) Find the best deterministic strategy and its competitive ratio (as a function of ).
(b) Show a tight lower bound for any deterministic algorithm for the problem.
2. Consider a variant of the paging problem in which at each page fault the algorithm is allowed to change
at most r pages in the memory for a cost of 1. Note that clearly r k which is the size of the memory.
(a) Find the best deterministic competitive algorithm for this variant (as a function of k and r)
Remark: note that also the optimal algorithm can change up to r pages in the memory for a cost
of 1.
(b) Design a matching lower bound for any deterministic algorithm for any k and r. Recall that the
online algorithm may replace up to r pages when there is a page fault at cost of 1.
(c) Design O(log k) randomized algorithm for r = 7. Hint: By how much at best the optimal cost can
be improved here.
3. Consider the online load balancing problem on (even) m related machines. The speed of each of the
