Divide and Conquer

• Somehow divide problem into $X$ spots of $n$ items, each size $n/X$
• Solve $X$ such problems
• Solutions of each combined into entire problem
• $T(n) = T(n/d) +$ some cost
• Master theorem

Merge Sort

• Select pivot, divide
• Merge, can do linearly
• $T(n) = 2T(n) = O(N\log N)$
• Note that we can find the median in linear time:
  • Pick randomly pivot X
  • Divide into two sets bigger and smaller than X
  • Note that now $X$ is in its correct position
  • Remove the smaller set, going in the direction of the pivot
  • Repeat with smaller array until found median

Deterministic median finding

$\Theta(n)$ time

(Idea median of medians)