Data Structures and Algorithms: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

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1Formulas and methods

Graph theory

Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij

Method

  1. Write down the degree of every vertex; the degrees add up to twice the number of edges.
  2. For Euler questions, check that the graph is connected and count the vertices with odd degree.
  3. For trees, use n − 1 edges and the handshake lemma.
  4. For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
  5. For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).

2Practice problems

1.Algorithms & complexityMedium

How does Prim's algorithm build a minimum spanning tree?

  1. AIt sorts all edges by weight and adds each one unless it would form a cycle in the forest
  2. BIt grows one tree from a start vertex, adding the cheapest edge that leaves the tree
  3. CIt removes the most expensive edge until no cycles remain
  4. DIt runs breadth-first search and keeps the BFS tree
2.Algorithms & complexityHard

Which two things must be shown to prove that a decision problem X is NP-complete?

  1. AX has no polynomial algorithm, and X is decidable
  2. BX is in NP, and a known NP-complete problem reduces to X in polynomial time
  3. CX reduces to a problem in P, and X is in NP
  4. DX is in P, and X reduces to SAT
3.Data structuresMedium

What is the cost of inserting an element in the middle of an array of n elements?

  1. AO(n²), because each element moves n times
  2. BO(n), because later elements must shift
  3. CO(1), because the index is known
  4. DO(log n), because of binary search
4.Data structuresHard

An object is stored as a key in a hash map, and then a field used by its hash function is changed. What is the likely result?

  1. ANothing changes, because a key's hash is computed only once when the program starts
  2. BThe map automatically moves the key to its new bucket
  3. CLooking up the object may fail because it sits in the bucket for its old hash
  4. DThe map throws an error at the moment the field changes
5.Graph theoryMedium

How many different Hamilton circuits H does the complete graph K6 have? Circuits that differ only in the starting vertex or the direction count as the same.

K6
H =
6.Graph theoryHard
12517206810ABCDE

Prim’s algorithm builds a minimum spanning tree starting from vertex D. Click the edge it adds as edge number 4.

start: Dk = 4
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge CE
  6. 6edge DE
  7. 7edge BE

3Solutions

1.Algorithms & complexity

Answer: B · It grows one tree from a start vertex, adding the cheapest edge that leaves the tree

Why this is the answer

Prim's algorithm, with a priority queue of crossing edges, runs in O(E log V) with a binary heap.

Answer

B · It grows one tree from a start vertex, adding the cheapest edge that leaves the tree
2.Algorithms & complexity

Answer: B · X is in NP, and a known NP-complete problem reduces to X in polynomial time

Why this is the answer

Membership in NP shows X is not harder than NP; the reduction from a known NP-complete problem shows X is at least as hard as all of NP.

Answer

B · X is in NP, and a known NP-complete problem reduces to X in polynomial time
3.Data structures

Answer: B · O(n), because later elements must shift

Why this is the answer

All elements after the insertion point move one position to the right, which takes time proportional to n.

Answer

B · O(n), because later elements must shift
4.Data structures

Answer: C · Looking up the object may fail because it sits in the bucket for its old hash

Why this is the answer

The key's location was chosen from its old hash; with a new hash, lookups search the wrong bucket, so keys should be immutable.

Answer

C · Looking up the object may fail because it sits in the bucket for its old hash
5.Graph theory

Answer: H = 60

Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.

Count

H = (6 − 1)!/2 = 120/2 = 60

Answer

H = 60
6.Graph theory

Answer: edge BE

Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.

Prim’s algorithm

1. DE (8)
2. CE (6)
3. AC (5)
4. BE (10)

edge number 4

BE

Answer

edge BE

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