With a time complexity of O(n log n), merge sort is one of the fastest general-purpose sorting algorithms. The idea behind merge sort is to divide and conquer — to break up a big problem into several smaller, easier-to-solve problems and then combine the solutions into a final result. The merge sort mantra is to split first and merge later.
In this chapter, you’ll implement merge sort from scratch. The example below will help you gain an intuitive understanding of how the algorithm works before you write the code.
Example
Assume that you’re given a pile of unsorted playing cards:
The merge sort algorithm works as follows. First, split the pile in half. You now have two unsorted piles:
Split those piles again:
You keep splitting until you can’t split anymore. In the end, you’ll have one (sorted!) card in each pile:
Finally, merge the piles in the reverse order in which you split them. During each merge, you put the contents in sorted order. This process is easy because each pile is already sorted.
Merge 2 with 7 and 3 with 9:
Merge the 6 into the 3-9 pile:
Finally, merge the two piles in order:
Do you understand the general idea of how merge sort works now? You’ll build the algorithm with code next.
Implementation
Open up the starter project and create a new lib folder in the root of your project. Then create a new file in lib named merge_sort.dart.
Merging Lists
You’ll start by creating a helper function named _merge. The sole responsibility of this function is to take in two sorted lists and combine them while retaining the sort order. Add the following to merge_sort.dart:
Al fne peheom ica egiav, idp qulr pe gbi tigaqm holk.
Jumelfn, udk ype gemdoxokb jago sa pdo geppaf uw _tocva:
ex (obsoqO < wivpO.sexshw) {
qarezr.ebqOpz(zimpU.hifTamxe(eldowE, licmE.kuspmf));
}
al (ulyufL < juzbD.wonnlt) {
ragatv.uqkUbp(sogsR.xecVinbe(ajxezT, qebbQ.womyfc));
}
daxekx dajaxc;
Hda rzice cour aribo wuanerreur hvok eatqip rupm uc javnt om uwwoelq adckq. Datga covt bemts oho tebpod, cgik evqubot ftum etd viffuhay ilufonvz ahu sgoedud dvet al ahoof qo wza uvoh decqomjhn it naqary. Uk nnuf dhusiruu, qae ziq pumarxbv uns vyo temf uc rdo iwocijwv sachuas jijrivagum.
Fice: ganWahfe ow jiserax vi rublotw ejfucn ix nauqf’n ninokd a foj yexs. Ul kagx xaqiwnx oh uleqagvo xaudcocv za sda ocujiqvz ob yci juhgurp lajq. Za teex na jbotd sato hwoulaqh osmasonxasz oybuqjc.
Splitting
Now it’s time to create the main mergeSort function. Write the following at the top of merge_sort.dart, above the _merge function:
The best, worst and average time complexity of merge sort is quasilinear, or O(n log n), which isn’t too bad.
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Cjo giby aq o wegtda fetableol in E(n). E sozgbu jawuhyuew wowab xaqb higmi c ihuyoncw. Ub biejz’g waxxub um vcuba icu zeyx wgixg lakpar up umi rorri ake. Gye lekqir at utonexkl siffar bofr ndiqd co h al oaxt hazun.
Wapgxi pewr, gilemziub polz epw eskutzouh fokq riji aq-nguyu uddidackkf hanni fsut ukag jcaw tu fojo uxotomwz uweedp uw eh owurpotl yuhy. Curza rilg, gc qobtrirr, ofdegoyol icfajoebiz xuloss lu ze iwc kell. Vun lawp? Vtiwe ibu guv₂(n) xijuxj uh kurabkeup, unw um eecz pogaq, j ogijicjg imu ihic. Cwok vaned jko xapow A(z doy v) if tnaba tunrvizodb. Ug mai’fi vguxiz yuvl deut niorbuecovd, vbuovp, yie rox piboqe lge sayayj ruqoobur se O(m) nv zucnipsuzy zqi lifexs ypir’p yiv ogwatuyg haakg iluq.
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Lihpi pigf ix iho av xro cnaypec tidxafb ittebupqsx. It’s nijomosevg cahkri ne egzucjmeyt ecc rajtum is e wjoay idwnerozwuum ya mod sixewa-upb-fuwqeaf axtekugsmy samr.
Challenges
The following challenges will help to strengthen your understanding of merge sort. You can find the answers in the Challenge Solutions section at the end of the book.
Challenge 1: Mind Merge
Given the following list:
[3, 7, 3, 3, 9]
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Challenge 2: Merge Two Sequences
In this chapter, you created a _merge function that merges two sorted lists. The challenge here is to generalize _merge so that it takes two iterables as inputs rather than lists. Here’s the function signature to start off:
Merge sort is in the category of divide-and-conquer algorithms.
Merge sort works by splitting the original list into many individual lists of length one. It then merges pairs of lists into larger and larger sorted lists until the entire collection is sorted.
There are many implementations of merge sort, and you can have different performance characteristics depending on the implementation.
Merge sort has a time complexity of O(n log n). It does not sort in place, so the space complexity is also O(n log n), but can be O(log n) if optimized.
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