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18th_oct_memoization
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18th_oct_memoization
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// memoization
// for example in fib so many duplicate computations (the tree)
// memoization : remember what we have computed earlier!
// memoization
// function records values that have previously been computed in local table
// we can use array as local table
function mfib(n){
const mem = [];
// serves as memory for already computed results of fib
function fib(k){
if (mem[k] !== undefined){
return mem[k]; // just access memory if already there
}
else {
const result =
k <= 1 ? k : fib(k-1) + fib(k-2);
mem[k] = result;
return result;
}
}
return fib(n);
}
// order of growth in time: theta(n)
// cause the duplicates no longer reevaluated
// using global memory instead:
// saving mem outside function
const mem = [];
function mfib_global(n){
if (mem[n] !== undefined){
return mem[n]; // just access memory if already there
}
else {
const result =
n <= 1 ? n : mfib_global(n-1) + mfib_global(n-2);
mem[n] = result;
return result;
}
}
// if we call mfib several times more efficient
// tribonacci sequences
// normal implementation:
function trib(n){
return n === 0
? 0
: n === 1
? 1
: n === 2
? 1
: trib(n-1) + trib(n-2) + trib(n-3);
}
// using memoization
const tribmem = []; // global memory
function mtrib(n){
if (tribmem[n] !== undefined) { // that is it is there in mem
return tribmem[n];
}
else {
const result =
n === 0
? 0
: n === 1
? 1
: n === 2
? 1
: mtrib(n-1) + mtrib(n-2) + mtrib(n-3);
tribmem[n] = result;
return result;
}
}
// Memoization abstract:
function memoize(f){
const mem = [];
function mf(x) {
if (mem[x] !== undefined){
return mem[x];
}
else {
const result = f(x);
mem[x] = result;
return result;
}
}
return mf;
}
const memo_trib = memoize(trib); // order of growth in time is still exponential
// cause we dont call back mf to access already calculated values
const memo_trib_better =
memoize(n => n === 0 ? 0
: n === 1 ? 1
: n === 2 ? 1
: memo_trib_better(n - 1) + memo_trib_better(n - 2) + memo_trib_better(n - 3));
// now we call back the original function...
// doubt: doesnt the mem[] become empty array again?
// but i think the memo_trib_better return mf... which is the inner function!
//memo_trib_better(23); // faster
//trib(23);
// order of growth: theta(n)
// n choose k
// normal implementation
function choose(n, k) {
return k > n
? 0
: k === 0 || k === n
? 1
: choose(n - 1, k) + choose(n - 1, k - 1);
}
// order of growth in running time : O(n^k)
// 2D arrays to store immediate results
// choose(n,k) in mem[n][k]
// read and write from/to Global 2-D Array
const mem_choose = [];
function read(n, k) { // read how many ways of choosing
return mem_choose[n] === undefined
? undefined
: mem_choose[n][k];
}
function write(n,k, value){
if (mem_choose[n] === undefined){
mem_choose[n] = [];
}
mem_choose[n][k] = value;
}
function mchoose(n,k){
if (read(n,k) !== undefined){
return read(n,k);
}
else {
const result = k > n ? 0
: k === 0 || k ==== n ? 1
: mchoose(n-1, k) +
mchoose(n-1, k-1);
write(n,k,result);
return result;
}
}
// order of growth in time:
// order of growth in space:
// can every tree recursive problems be memoized?