Forensics and CS Philip Chan

CSI: Crime Scene Investigation high tech forensics tools DNA profiling Use as evidence in court cases

DNA Deoxyribonucleic Acid Each person is unique in DNA (except for twins) DNA samples can be collected at crime scenes About.1% of human DNA varies from person to person

Forensics Analysis Focus on loci (locations) of the DNA Values at the those loci (DNA profile) are recorded for comparing DNA samples. Two DNA profiles from the same person have matching values at all loci. More or fewer loci are more accurate in identification? Tradeoffs? FBI uses 13 core loci

We do not want to wrongly accuse someone How can we find out how likely another person has the same DNA profile? How many people are in the world? How low the probability needs to be so that a DNA profile is unique in the world? Low probability doesn’t mean impossible Just very unlikely

Review of basic probability Joint probability of two independent events P(A,B) = ?

Review of basic probability Joint probability of two independent events P(A,B) = P(A) * P(B) Independent events mean knowing one event does not provide information about the other events P(Die1=1, Die2=1) = P(Die1=1) * P(Die2=1) = 1/6 * 1/6 = 1/36.

Enumerating the events ,11,2… events, each is equally likely, so 1/36

Joint probability P(Die1=even, Die2=6) = ?

Joint probability P(Die1=even, Die2=6) = 1/2 * 1/6 = 1/12 P(Die1=1, Die2=5, Die3=4) = ?

Joint probability P(Die1=even, Die2=6) = 1/2 * 1/6 = 1/12 P(Die1=1, Die2=5, Die3=4) = (1/6) 3 = 1/216

DNA profile probability How to estimate?

DNA profile probability How to estimate? Assuming loci are independent P(Locus1=value1, Locus2=value2,...) = P(Locus1=value1) * P(Locus2=value2) *...

DNA profile probability How to estimate? Assuming loci are independent P(Locus1=value1, Locus2=value2,...) = P(Locus1=value1) * P(Locus2=value2) *... How to estimate P(Locus1=value1)?

DNA profile probability How to estimate? Assuming loci are independent P(Locus1=value1, Locus2=value2,...) = P(Locus1=value1) * P(Locus2=value2) *... How to estimate P(Locus1=value1)? a random sample of size N from the population and find out how many people out of N have value1 at Locus1

Database of DNA profiles IdLocus1Locus2Locus3…Locus13 A5212 A6921 …

Problem Formulation Given A sample profile (e.g. collected from the crime scene) A database of known profiles Find The probability of the sample profile if it matches a known profile in the database

Breaking Down the Problem Find The probability of the sample profile if it matches a known profile in the database What are the subproblems?

Breaking Down the Problem Find The probability of the sample profile if it matches a known profile in the database What are the subproblems? Subproblem 1 Find whether the sample profile matches  1a: ?  1b: ? Subproblem 2 Calculate the probability of the profile

Breaking Down the Problem Find The probability of the sample profile if it matches a known profile in the database What are the subproblems? Subproblem 1 Find whether the sample profile matches  1a: check entries in the database  1b: check if all 13 loci match in each entry Subproblem 2 Calculate the probability of the profile

Simpler Problem for 1a (very common) Given an array of integers (e.g. student IDs) an integer (e.g. an ID) Find whether the integer is in the array int[] directory; // student id’s int id; // to be found boolean found; // true if id is in directory

Linear/Sequential Search Check one by one Stop if you find it Stop if you run out of items to check Not found

Number of Checks (speed of algorithm) Consider N items in the array Best-case scenario When does it occur? How many checks?

Number of Checks (speed of algorithm) Consider N items in the array Best-case scenario When does it occur? How many checks? First item;1 check Worst-case scenario When does it occur? How many checks?

Number of Checks (speed of algorithm) Consider N items in the array Best-case scenario When does it occur? How many checks? First item;1 check Worst-case scenario When does it occur? How many checks? Last item or not there; N checks Average-case scenario Average of all cases ( … + N) / N =

Can we do better? Faster algorithm? What if the array is sorted, items are in an order E.g. a phone book

Binary Search Check the item at midpoint If found, done Otherwise, eliminate half and repeat

Breaking down the problem While more items and not found What are the two subproblems?

Breaking down the problem While more items and not found Eliminate half of the items Find the mid point

Number of checks (Speed of algorithm) Best-case scenario When does it occur? How many checks?

Number of checks (Speed of algorithm) Best-case scenario When does it occur? How many checks? In the middle; 1 check

Number of checks (Speed of algorithm) Best-case scenario When does it occur? How many checks? In the middle; 1 check Worst-case scenario When does it occur? How many checks?

Number of checks (Speed of algorithm) Best-case scenario When does it occur? How many checks? In the middle; 1 check Worst-case scenario When does it occur? How many checks? Dividing into two halves, half has only one item ? checks

Number of checks (Speed of algorithm) Best-case scenario When does it occur? How many checks? In the middle; 1 check Worst-case scenario When does it occur? How many checks? Dividing into two halves, half has only one item ? checks

Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1

Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1

Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1

Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1

Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1 = [ [ T(N/8) + 1] + 1] + 1

Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1 = [ [ T(N/8) + 1] + 1] + 1 = … any pattern?

Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1 = [ [ T(N/8) + 1] + 1] + 1 = … = T(N/2 k ) + k

Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1 = [ [ T(N/8) + 1] + 1] + 1 = … = T(N/2 k ) + k N/2 k gets smaller and eventually becomes 1

Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1 = [ [ T(N/8) + 1] + 1] + 1 = … = T(N/2 k ) + k N/2 k gets smaller and eventually becomes 1 solve for k

Number of Checks (Speed of Algorithm) N/2 k = 1 N = 2 k k = ?

Number of Checks (Speed of Algorithm) N/2 k = 1 N = 2 k k = log 2 N

Number of Checks (Speed of Algorithm) N/2 k = 1 N = 2 k k = log 2 N T(N) = T(N/2 k ) + k = T(1) + log 2 N = ? + log 2 N

Number of Checks (Speed of Algorithm) N/2 k = 1 N = 2 k k = log 2 N T(N) = T(N/2 k ) + k = T(1) + log 2 N = 1 + log 2 N

Sorting (arranging the items in a desired order) How is the phone book arranged? Why? Why not arranged by numbers?

Sorting (arranging the items in a desired order) How is the phone book arranged? Why? Why not arranged by numbers? Order Alphabetical Low to high numbers DNA profile with 13 loci?

Sorting Imagine you have a thousand numbers in an array How would you systemically sort them?

Selection Sort (ascending) Find/select the smallest item Swap the smallest item with the first item

Selection Sort (ascending) Find the smallest item Swap the smallest item with the first item Find/select the second smallest item Swap the second smallest item with the second item …

Breaking down the problem Get all the items in ascending order Get one item at the wanted position/index What are the two subproblems?

Breaking down the problem Get all the items in ascending order Get one item at the wanted position/index Find the smallest item

Breaking down the problem Get all the items in ascending order Get one item at the wanted position/index Find the smallest item Swap the smallest item with the item at the wanted position

Number of comparisons (Speed of Algorithm) Consider counting Number of comparisons between array items

Number of comparisons (Speed of Algorithm) Consider counting Number of comparisons between array items Best-case scenario (least # of comparisons) When does it occur? How many comparisons?

Number of comparisons (Speed of Algorithm) Consider counting Number of comparisons between array items Best-case scenario (least # of comparisons) When does it occur? How many comparisons? Worst-case scenario (most # of comparisons) When does it occur? How many comparisons?

Number of comparisons (Speed of Algorithm) Consider counting Number of comparisons between array items Best-case scenario (least # of comparisons) When does it occur? How many comparisons? Worst-case scenario (most # of comparisons) When does it occur? How many comparisons? Same number of comparisons For all cases (ie best case = worst case)

Number of comparisons (Speed of Algorithm) To find the smallest item How many comparisons?

Number of comparisons (Speed of Algorithm) To find the smallest item How many comparisons? N-1 To find the second smallest item How many comparisons?

Number of comparisons (Speed of Algorithm) To find the smallest item How many comparisons? N-1 To find the second smallest item How many comparisons? N-2 … Total # of comparisons?

Number of comparisons (Speed of Algorithm) To find the smallest item How many comparisons? N-1 To find the second smallest item How many comparisons? N-2 … Total # of comparisons (N-1) + (N-2) + … + 1

Number of comparisons (Speed of Algorithm) To find the smallest item How many comparisons? N-1 To find the second smallest item How many comparisons? N-2 … Total # of comparisons (N-1) + (N-2) + … + 1 N(N-1)/2 = (N 2 – N)/2

Summary DNA samples from crime scene Identify people using known DNA profiles If there is a match estimate probability of DNA profile Matching a sample to known DNA profiles Linear/sequential search [N checks] Binary search [log 2 N + 1 checks] Faster but needs sorted data/profiles  Selection Sort [(N 2 – N)/2 comparisons]