HW6.pdf - Homework 6: Due November 18th 12pm ORIE 4520:...

ORIE 4520: Stochastics at Scale
Fall 2015
Homework 6: Due November 18th, 12pm
Sid Banerjee ( [email protected] )
Problem 1: (The Flajolet-Martin Counter)
In class (and in the prelim!), we looked at an idealized algorithm for finding the number of distinct
elements in a stream, where we sampled uniform random variables for each item, and then stored
their minimum value. One way to implement this in practice is via the Flajolet-Martin counter:
Suppose we have a stream ( X
1
, X
2
, . . . , X
m
) of m items, where each item X
i
corresponds to an
element in [ n ]. Assume n is a power of 2, and k = log
2
n . Let h be a hash function that maps each
of the elements in [ n ] to k bits – in particular, let us denote h ( x ) = ( b
1
( x ) , b
2
( x ) , . . . , b
k
( x )) for each
x ∈ [ n ]), and assume that each bit k independently satisfies P [ b
k
( x ) = 0] = P [ b
k
( x ) = 1] = 1 / 2
for every pair x ∈ [ n ]. For every x ∈ [ n ], let r ( x ) be the number of trailing 0 ’s in h ( x ) – so for
example, for n = 16 (i.e., k = 4), h ( x ) = 0100 means r ( x ) = 2, h ( x ) = 1000 means r ( x ) = 3, and
so on). Finally, let R = max
i
{ r ( X
i
) } – i.e., the maximum number of trailing 0’s in the hashes of
the items in the stream.
Part (a)
For any element x ∈ [ n ], let Y
j
( x ) be the indicator that r ( x ) = j . Argue that E [ Y
j
( x )] = 1 / 2
j +1
.
Part (b)
Let F
0
be the number of distinct elements in the stream, and define N
j
to be the number of elements
in the stream for which r ( x ) > j . Show that:
E [ N
j
] =
F
0
2 j +1
, V ar ( N
j
) =
F
0
2 j +1
1 -
1
2 j +1
≤ E [ N
j
]
Hint: Write N
j
in terms of Y
j
( X
i
) .
Part (c)
Suppose we use 2
R
as an estimator for F
0
. Argue that for any j , P [ R ≥ j ] ≥ P [ N
j
> 0]. Next,
assuming that F
0
is a power of 2, show that:
P [ R < log
2
( F
0
) - c ] ≤
1
2 c
Hint: Use Chebyshev to bound P [ N
j
= 0] .
Part (d)
On the other hand, argue that P [ R ≥ j ] ≤
∑
j 0 ≥ j
E [ N
j
0
], and hence show that:
P [ R ≥ log
2
( F
0
) + c ] ≤
1
2 c
1

ORIE 4520: Stochastics at Scale
Fall 2015
Homework 6: Due November 18th, 12pm
Sid Banerjee ( [email protected] )
Problem 2: (An Alternate All-Pair Distance Sketch)
In class we saw an All-Pairs Distance Sketch (ADS) by Das-Sarma et al., which for each node v
stored a sketch S ( v ) with distances to O (log n ) other nodes, and then given any pair ( u, v ), used
the to sketches to compute a shortest-path estimate within a multiplicative ‘stretch’ of O (log n ).
We’ll now see an alternate ADS proposed by Cohen et al.
We are given an undirected weighted graph G ( V, E ), where each edge ( u, v ) ∈ E has some weight
w
u,v
≥ 0 corresponding to its length. The shortest path distance d
*
( u, v ) between any nodes u and v
is the minimum sum of weights over all paths from u to v . For convenience, assume that the weights