CS 1510 Adversarial Lower Bound Arguments
1. In a simplified form of the game mastermind there is a hidden sequence H= (c1,...,ck) of kcolored
pegs. There are Cdifferent possible colors. Colors can be repeated in the hidden sequence. The game
consistes of repeated rounds. To start a round the guesser gives the hider a sequece G= (g1,...,gk)
of kcolors. The hider then tells the guesser how many of the guesses were correct, that is, the number
of indices jsuch that hj=gj. This ends a round.
Compute a lower bound as a function of kand Con the number of the number rounds required by the
guesser to guarantee that he/she will solve the puzzle. A typical values of k= 4 and C= 6; what do
you get for your lower bound in this case.
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2. In the game mastermind there is a hidden sequence H= (c1,...,ck) of kcolored pegs. There are
Cdifferent possible colors. Colors can be repeated in the hidden sequence. The game consistes of
repeated rounds. To start a round the guesser gives the hider a sequece G= (g1,...,gk) of kcolors.
The hider then tells the guesser how many of the guesses were correct, that is, the number of indices
jsuch that hj=gj. In addition, the hider tells the guess how many colors are correct, but are in the
wrong position (think of Hand Gas being multi-sets and the hider tells the guesser the cardinality of
the multi-set intersection of Hand G). This ends a round.
Compute a lower bound as a function of kand Con the number of the number rounds required by the
guesser to guarantee that he/she will solve the puzzle. A typical values of k= 4 and C= 6; what do
you get for your lower bound in this case.
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3. Prove that computing the OR of nbits requires Ω(log n) steps on a EREW PRAM, independent of
the number of processors used.
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4. Consider the following situation. You have two workstations Aand Bconnected by a communication
line. The workstation Ais initially given an nbit integer x. The workstation Bis initially given an
nbit integer y. The goal of the two workstations is to communicate over the line in such a way that
they both know the number of bits that are 1 in xplus the number of bits that are 1 in y. Show that
the number of bits sent a cross the line must be Ω(l og n).
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5. Consider the following situation. You have two workstations Aand Bconnected by a communication
line. The workstation Ais initially given an nbit integer x. The workstation Bis initially given an
nbit integer y. The goal of the two workstations is to communicate over the line in such a way that
they both know the bit-wise exclusive-or of xand y. Show that the number of bits sent across the line
must be Ω(n).
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6. You goal is to find a counterfeit coin among a group of 9 coins. Counterfeit coins are lighter than
real coins. You know that exactly one of the 9 coins is counterfeit. To help you decide which coin is
legitimate you have a pan balance. A pan balance functions in the following manner. You can give
the pan balance any two disjoint subcollections, say S1and S2, of the coins. Let |S1|and |S2|be the
cumulative weight of the coins in S1and S2, respectively. The pan balance then determines whether
|S1|<|S2|,|S1|=|S2|, or |S1|>|S2|. Show how to solve this problem in two weighings on the pan
balance.
HINT: This is not a lower bound problem. It’s purpose is to build you intution.
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