Download Linear Time Selection: Minimizing Construction Cost of Building Driveways and more Slides Computer Science in PDF only on Docsity!
Linear Time Selection
Where to Build a Road?
Given (x,y) coordinates of N houses, where should you build roadparallel to x-axis to minimize construction cost of building driveways?
2
9
4
5
8
7
6
3
11
12
10
1
3
Where to Build a Road?
Given (x,y) coordinates of N houses, where should you build roadparallel to x-axis to minimize construction cost of building driveways?
n
n
1
= nodes on top.
n
n
2
= nodes on bottom.
2
9
4
5
8
7
1
6
3
11
Decreases total cost by (n
2
-n
1
ε
12
10
Where to Build a Road?
Given (x,y) coordinates of N houses, where should you build roadparallel to x-axis to minimize construction cost of building driveways?
n
n
1
= nodes on top.
n
n
2
= nodes on bottom.
2
9
4
5
8
7
1
6
3
11
12
10
Docsity.com
5
Where to Build a Road?
Given (x,y) coordinates of N houses, where should you build roadparallel to x-axis to minimize construction cost of building driveways?
n
n
1
= nodes on top.
n
n
2
= nodes on bottom.
2
9
4
5
8
7
1
6
3
11
12
10
Where to Build a Road?
Given (x,y) coordinates of N houses, where should you build roadparallel to x-axis to minimize construction cost of building driveways?Solution: put street at median of y coordinates.
2
9
4
5
8
7
1
6
3
11
12
10
7
Order Statistics
Given N linearly ordered elements, find i
th
smallest element.
n
Min:
i = 1
n
Max: i = N
n
Median: i =
(N+1) / 2
and
(N+1) / 2
n
O(N) for min or max.
n
O(N log N) comparisons by sorting.
n
O(N log i) with heaps.
Can we do in worst-case O(N) comparisons?
n
Surprisingly, yes. (Blum, Floyd, Pratt, Rivest, Tarjan, 1973)
Assumption to make presentation cleaner.
n
All items have distinct values.
Select
Similar to quicksort, but throw away useless "half" at each iteration.
n
Select i
th
smallest element from a
1
, a
2
,... , a
N
x
Partition(N, a
1
, a
2
, ..., a
N
k
rank(x)
if (i == k)
return x
else if (i < k)
b[]
all items of a[] less than x
return Select(i
th
, k-1, b
1
, b
2
, ..., b
k-
else if (i > k)
c[]
all items of a[] greater than x
return Select((i-k)
th
, N-k, c
1
, c
2
, ..., c
N-k
Select (i
th
, N, a
1
, a
2
,... , a
N
x = partition element
Want to choose x so thatx is (roughly) the ith largest.
Docsity.com
13
median ofmedians
Selection Analysis
Crux of proof: delete roughly 30% of elements by partitioning.
n
At least 1/2 of 5 element medians
x
at least
N / 5
N / 10
medians
x
median ofmedians
Selection Analysis
Crux of proof: delete roughly 30% of elements by partitioning.
n
At least 1/2 of 5 element medians
x
at least
N / 5
N / 10
medians
x
n
At least 3
N / 10
elements
x.
15
Selection Analysis
Crux of proof: delete roughly 30% of elements by partitioning.
n
At least 1/2 of 5 element medians
x
at least
N / 5
N / 10
medians
x
n
At least 3
N / 10
elements
x.
n
At least 3
N / 10
elements
x.
median ofmedians
Selection Analysis
Crux of proof: delete roughly 30% of elements by partitioning.
n
At least 1/2 of 5 element medians
x
at least
N / 5
N / 10
medians
x
n
At least 3
N / 10
elements
x.
n
At least 3
N / 10
elements
x.
Select()
called recursively (Case 2 or 3) with at most
N - 3
N / 10
elements.
C(N) = # comparisons on a file of size N.Now, solve recurrence.
n
Apply master theorem?
n
Assume N is a power of 2?
n
Assume C(N) is monotone non-decreasing?
R
Q
R
P I R R R R
Q
R
R
R
R
P
I
RQ
RP
I
sort
insertion
select
recursive
medians
of
median
N O N N C N C N C
≤
Docsity.com
17
Selection Analysis
Analysis of selection recurrence.
n
T(N) = # comparisons on a file of size
N.
n
T(N) is monotone, but C(N) is not!
Claim: T(N)
20cN.
n
Base case: N < 50.
n
Inductive hypothesis: assume true for 1, 2,... , N-1.
n
Induction step: for N
50, we have:
otherwise
if
T(
cN
N
N
T
N
T
N
cN
N
cN
cN
N c N c N c
cN
N
N
c
N
c
cN
N N T N T N T
For n
N / 10
N / 4.
Linear Time Selection Postmortem
Practical considerations.
n
Constant (currently) too large to be useful.
n
Practical variant: choose random partition element.
O(N) expected running time ala quicksort.
n
Open problem: guaranteed O(N) with better constant.
Quicksort.
n
Worst case O(N log N) if always partition on median.
n
Justifies practical variants: median-of-3, median-of-5.
Docsity.com
