最小生成树
- Algorithm
- - Prim
  - - Code
  - Kruskal
- Prim&KruskalPrim \& KruskalPrim&Kruskal算法的性质
- 最小生成树的性质
- - 最小生成树的线性递增序列
- 例题
次小生成树
- Code
- 例题
- - 1148. 秘密的牛奶运输
  - 356. 次小生成树

最小生成树

Given a Connected-Undirected-Graph with nnn points (it may contains Negative-Edge, Self-Loop, Negative-Loop, in other words, a arbitrary graph but satisfying Connected-Undirected), you need to find a Sub-Graph which contains nnn points, being connected and its weight (the sum of all edges) is minimal amongst others.

In fact, such a Sub-Graph could be a Tree (n−1n - 1n−1 edges).
Because there may have many different Sub-Graph which share the same weight, we assert that there must exist a Tree amongst them.

There are some algorithms for getting a MST.

Algorithm

Prim

We have a tree TTT with a single arbitrary point originally, each time we add a point and a edge (which connects this new point and a point of TTT), after n−1n - 1n−1 times we will get the MST.

We stipulate AdjEdges(T)AdjEdges( T)AdjEdges(T) is the set of all adjacent edges which connect a point of TTT and a point that is not belongs to TTT.

For example, the MST is

         {b,...}|
{c,...}-----a|{d,...}+ the edges {d-c, d-f, d-e} are belong to MST.
+ {b,...} means a `Sub-Tree` of the MST, the same as {c,...} and {d,...}

Original, we set T=(a)T = (a)T=(a)

First time, (suppose that the weight of a−ba-ba−b is less than that of a−ca-ca−c and a−da-da−d), our goal is to get a−ba-ba−b, and then add it to TTT.

Property: a−ba-ba−b is the minimal edge of AdjEdges(T)AdjEdges( T)AdjEdges(T)
Because a−ba-ba−b is less than a−ca-ca−c and a−da-da−d as we assumed, so we need to prove a−ba-ba−b is less than any edge of AdjEdges(T)/{a−b,a−c,a−c}AdjEdges( T) / \{a-b, a-c, a-c\}AdjEdges(T)/{a−b,a−c,a−c}

Proof:

T: {a}------{b,...}|------{c,...}|------{d,...}

if there exists a edge a−Ba-Ba−B (B∈{b,...}B \in \{b,...\}B∈{b,...}) which is less than a−ba-ba−b, we can replace a−ba-ba−b by a−Ba-Ba−B to get a new-MST, however it has a smaller weight than the original-MST, so it raises a contradiction.
If there exists a edge a−Ca-Ca−C (C∈{c,...}C \in \{c,...\}C∈{c,...}) which is less than a−ba-ba−b, we can also replace a−ca-ca−c by a−Ca-Ca−C to get a smaller new-MST which also makes a contradiction.

More generally, if TTT contains multiple points, e.g., {a,x,y}\{a, x, y\}{a,x,y}. Now, the edge ?−b?-b?−b means there is a edge between ??? (a point of TTT) and bbb in the MST.
Supposing ?−b?-b?−b is less than ?−c,?−d?-c, ?-d?−c,?−d, and you can also prove that ?−b?-b?−b is the minimal edge of AdjEdge(T)AdjEdge( T)AdjEdge(T).

Therefore, the process of Prim is consisted of nnn steps:

Firstly, T=(a)T = (a)T=(a) (aaa is a arbitrary point)
Secondly, suppose that x−yx-yx−y is the minimal edge of AdjEdges(T)AdjEdges( T)AdjEdges(T) (xxx belongs to TTT), then we add yyy to TTT and x−yx-yx−y is a edge of MST.
The same as Secondly.

Code

We will use a auxiliary array Distance to finish this algorithm.

Assume that T=(a,b,c)T = (a, b, c)T=(a,b,c) at current and others undetermined points are (d,e,f)(d, e, f)(d,e,f), then Distance[d]=min(Edge[d−a],Edge[d−b],Edge[d−c])Distance[ d] = min( Edge[d-a], Edge[d-b], Edge[d-c])Distance[d]=min(Edge[d−a],Edge[d−b],Edge[d−c]), that is, the DistanceDistanceDistance of a undetermined point xxx is the minimal edge of all edges which connect xxx and a point of TTT.

Next, we iterate all undetermined points (d, e, f), find the minimal DistanceDistanceDistance (suppose it is ddd), then ddd is added to TTT, and Distance[d]Distance[d]Distance[d] is a new edge of MST.

Finally, because ddd is belongs to TTT, so Distance[e]=min(Edge[e−a],Edge[e−b],Edge[e−c])Distance[ e] = min( Edge[e-a], Edge[e-b], Edge[e-c])Distance[e]=min(Edge[e−a],Edge[e−b],Edge[e−c]) should also be updated by Edge[e−d]Edge[e-d]Edge[e−d].
… that is, iterate all undetermined points (e,f), update Distance[x]=min(self,Edge[x,d])Distance[ x] = min( self, Edge[ x, d])Distance[x]=min(self,Edge[x,d]). (ddd is the new point that added into MST).

Kruskal

Auxiliary-Property-1:
For a graph, its MST must contains eee which is the minimal edge of the graph.
(proof: e.g., a--b--c is a branch of MST, and there exists a edge a-c that is the minimal edge amongst all edges of the graph; then you can replace a--b by a-c to make the MST more optimal)

Auxiliary-Property-2:
For a graph (contains (a,b,c,d,e)(a,b,c,d,e)(a,b,c,d,e) points), when we divide its MST into two partition, e.g., if c−dc-dc−d is a edge of MST, then remove this edge, we will get two Sub-Tree of MST (denote M1,M2M1, M2M1,M2), let it be (a,b,c)(a,b,c)(a,b,c) and (d,e)(d,e)(d,e). And also divide this graph into two partition (denote G1,G2G1, G2G1,G2): one contains (a,b,c)(a,b,c)(a,b,c) points and its interior edges, another contains (d,e)(d,e)(d,e) points and its interior edges.
Then, M1M1M1 is also the MST of G1G1G1, the same as M2,G2M2, G2M2,G2.
(proof: if mmm is the MST of G1G1G1 instead of M1M1M1, then the Tree m,(c−d),m2m, (c-d), m2m,(c−d),m2 is also a MST which is less than the previous MST, this is contradictory.)

Originally, let edgesedgesedges be all edges of the graph. Stipulate min(edges)min( edges)min(edges) denotes the minimal-edge of edgesedgesedges.

min(edges)min( edges)min(edges) is a edge (let its endpoints be a,ba,ba,b) of MST (according to Property-1), and divide the MST into two partition M(a),M(b)M(a), M(b)M(a),M(b);
remove all edges that connect M(a),M(b)M(a), M(b)M(a),M(b) from edgesedgesedges, it also means you divide the original-Graph into two parts G(a),G(b)G(a), G(b)G(a),G(b) (edgesedgesedges is the edges that either belongs G(a)G(a)G(a) or belongs to G(b)G(b)G(b))
According to Property-2, M(a)M(a)M(a) is also the MST of G(a)G(a)G(a)
suppose min(edges)min(edges)min(edges) belongs to G(a)G(a)G(a), then min(edges)min(edges)min(edges) is a edge of M(a)M(a)M(a). Using the same process as the above, you will get 333 Sub-Graphs.
Until you get nnn Sub-Graphs (each graph is a point), then the n−1n-1n−1 edges which are removed is the MST.

As a result, sort all edges at first, and iterate each edge (a,b,w)(a,b,w)(a,b,w), if a,ba, ba,b are not in a same Disjoint-Set, then www is a edge of MST (the removal-edge as mentioned above), then Merge a,ba, ba,b to the same Set.
Otherwise a,ba,ba,b is in the same Set, meaning the operation that remove a edge that connects two divided-Graph (G(a),G(b)G(a), G(b)G(a),G(b) as mentioned above)

Prim&KruskalPrim \& KruskalPrim&Kruskal算法的性质

+ The edges of the graph in Prim or Kruskal, are both undirected (bidirectional).

+ If the graph is not connected, that is consists of several connected-subgraphs G0,Gi,...G0, Gi, ...G0,Gi,... (where G0G0G0 denotes a connected-subgraph that contains point-000), Prim can only get the MST of G0G0G0 not for Gi,...Gi, ...Gi,...; while Kruskal could get all MSTs of G0,Gi,...G0, Gi, ...G0,Gi,... simultaneously.

+ Prim, Kruskal can also be used to calculate Maximal-ST, by replacing minminmin by maxmaxmax.
. Proof: ???

+ For a undirected-graph GGG, when you storing it
. In Prim, for any undirected-edge (a,b,w) (assume there is no other edges between (a,b)), you must make sure Edges[a][b] = Edges[b][a] = w in the Adjacency-Matrix. That is, one undirected-edge corresponds to two directed-edges.
. In Kruskal, you can just choose one directed-edge (a->b) to be stored in Linked-List, not necessary both sides; this is due to the mechanism of Kruskal (the disjoint-set not relates the direction of edges) Consequently, one undirected-edge just corresponds to its one directed-edge is enough.

最小生成树的性质

最小生成树的线性递增序列

We call a Sequence of a ST is a ordered sequence of edges after sorting all edges;
. 1 For any one MST, if its Sequence is a1≤a2≤...a1 \leq a2 \leq ...a1≤a2≤..., then the Sequences of all MSTs are identical to a1,a2,...a1, a2, ...a1,a2,...
. 2 More generally, for any ST, if its Sequence is b1≤b2≤...b1 \leq b2 \leq ...b1≤b2≤..., then we have bi≥aibi \geq aibi≥ai
. For instance, if a MST is 1 2 4, then there would not exist a ST (maybe MST) which is 1 1 5 or 3 3 3, due to bi≥aibi \geq aibi≥ai. But 1 3 4 (non-MST) is possible.
. This property is very important and useful, it not only shows that the sum of all edges in a MST is ≤\leq≤ that of any ST, but also implies that the maximal edge in a MST must ≤\leq≤ any ST (maybe equal, e.g., the MST is 1 2 4, a non-MST ST is 1 3 4, such cases are possible).

例题

+ 1144. 连接格点

次小生成树

Let MST be the edges e1,e2,...eme1, e2, ...eme1,e2,...em, then all inferior-MST (strictly, or non-strictly) must be the edges f1,f2,...,fmf1, f2, ..., fmf1,f2,...,fm where only one pair of edges ei≠fjei \neq fjei=fj, in other words, if we remove ei,fjei, fjei,fj, the rest two sets will form a bijection.

Let the context of the next discussion be Strictly Inferior-MST.

If we have a MST e1,e2,...,eme1, e2, ..., eme1,e2,...,em, then we iterator all edges on GGG which are not eieiei (i.e., the edges not in the MST)
For any such a edge fififi (if its endpoints are a,ba, ba,b, and the path of them in the MST are ex,ey,ezex, ey, ezex,ey,ez, the weight are 3,5,53, 5, 53,5,5)
There is a necessity: fi≥5fi \geq 5fi≥5 (if not, that is fi<5fi < 5fi<5, we move 555 from MST and add fififi, it again forms a ST whose sum less than MST;) this is a property of MST, which is no matter to inferior-MST.
. According to the property above, if the inferior-MST contains fififi, then the edges of inferior-MST must be obtained by the process: remove a edge ex/ey/ezex/ey/ezex/ey/ez from MST, and then add fififi
There are also some lemmas on the relationship between the removal edge and fififi.
1 when fi>5fi > 5fi>5, only if fififi replace the maximal edge 555, that could maybe the inferior-MST (because if fififi replace 1/31/31/3, it will get a much large ST than replacing 555)
2 otherwise fi=5fi = 5fi=5, if the inferior-MST is strictly, then we shall try to use fififi to replace the strictly inferior-maximal edge 333 (although there are multiple same maximal edges 555; e.g., the MST is 3,5,53, 5, 53,5,5 and there is only one edge 555 which not belongs to MST, so all STs with distinct sum are 3,5,53,5,53,5,5 and 5,5,55,5,55,5,5; that is, now we should choose the strictly inferior-maximal-edge 333 on the MST-path, not the non-strictly 555)

That is, we need get d1d1d1 means the maximal edge of the path a−ba - ba−b in MST, d2d2d2 means the strictly inferior-maximal edge in that path (note that, if the path is 3,5,53, 5, 53,5,5, then d1=5,d2=3d1 = 5, d2 = 3d1=5,d2=3, that is the case of strictly inferior-MST)

Code

Suppose all edges are ≥0\geq 0≥0 and we use −1-1−1 denotes a invalid-edge;

Strictly-Inferior-MST

//> get the maximal and inferior-maximal edges on the path (a-b) in the MST
pair< int, int> Get_edges( int a, int b){d1 = d2 = -1; //< d1 is the maximal-edge, d2 is the inferiore-maximalfor( auto w : Edges on the path `a-b` in MST){update( d1, d2, w);}
}void update( int & d1, int & d2, int _w){if( _w > d1){d2 = d1;d1 = _w;}else if( _w != d1){ //< d2 must `<` d1d2 = max( _w, d2);}
};
//----//> find the Strictly-Inferior-MST
ans = 0x7F7F7F7F...;
for( edge `(a-b):w` : all edges of the graph ){if( this edge on the MST){continue;}if( a == b){continue;}pair<int,int> ret = Get_edges( a, b);ASSERT_( ret.first != -1);if( w > ret.first){ans = min( mst_size - ret.first + w, ans);}else if( ret.second != -1){ans = min( mst_size - ret.second + w, ans);}
}

Non-Strictly-Inferior-MST

//> get the maximal edge on the path (a-b) in the MST
pair< int, int> Get_edges( int a, int b){d1 = -1; //< d1 is the maximal-edgefor( auto w : Edges on the path `a-b` in MST){d1 = max( d1, w);}
}
//----//> find the NonStrictly-Inferior-MST
ans = 0x7F7F7F7F...;
for( edge `(a-b):w` : all edges of the graph ){if( this edge on the MST){continue;}if( a == b){continue;}auto ret = Get_edges( a, b);ASSERT_( ret != -1);ans = min( mst_size - ret + w, ans);
}

例题

1148. 秘密的牛奶运输

problem-link

Cuz there are only N=500N = 500N=500 points, when we got the MST, we can use D1[N][N],D2[N][N]D1[N][N], D2[N][N]D1[N][N],D2[N][N] denotes the Maximal and Inferior-Maximal edge on the path (a−b)(a-b)(a−b) in the MST. More precisely, let Dfs(x)Dfs(x)Dfs(x) to get all D1[x][all],D2[x][all]D1[x][all], D2[x][all]D1[x][all],D2[x][all], then performing Dfs(all)Dfs( all)Dfs(all)

356. 次小生成树

link

Due to N=1e5N = 1e5N=1e5, the above N2N^2N2 device is infeasible here.
Cuz MST is a tree, we can use LCA algorithm which will divide a path a−ba-ba−b into several disjoint segments (is also a path essentially) s1,s2,...s1, s2, ...s1,s2,..., then we can store d1,d2d1, d2d1,d2 which represent the maximal and inferior-maximal edge for each segment sisisi, then we gather them up (d1,d2)(d1,d2)...(d1, d2) (d1, d2) ...(d1,d2)(d1,d2)... and the maximal and inferior-maximal edge on the path a−ba-ba−b must be contained in this set.

//> building `dist1, dist2`
for( int power = 0; power < power_range; ++power){if( power == 0){ptrNew_father[ nex][ power] = cur;dist1[ nex][ power] = wth;dist2[ nex][ power] = -1;continue;}int mid = ptrNew_father[ nex][ power - 1];if( (mid == -1) || (ptrNew_father[ mid][ power - 1] == -1)){ptrNew_father[ nex][ power] = -1;dist1[ nex][ power] = -1;dist2[ nex][ power] = -1;continue;}ptrNew_father[ nex][ power] = ptrNew_father[ mid][ power - 1];int _edges[ 4] = { dist1[ nex][ power - 1], dist2[ nex][ power - 1],dist1[ mid][ power - 1], dist2[ nex][ power - 1]};sort( _edges, _edges + 4);dist1[ nex][ power] = _edges[ 3];dist2[ nex][ power] = -1;for( int i = 2; i >= 0; --i){if( _edges[ i] != dist1[ nex][ power]){dist2[ nex][ power] = _edges[ i];break;}}ASSERT_( dist2[ nex][ power] < dist1[ nex][ power]);}
//----
//> get the maximal and inferior-maximal edges on the path (a-b) in MST.
pair< int, int> Get_edges( int _a, int _b){int d1 = -1, d2 = -1;ASSERT_( (_a >= 0) && (_a < ptrRef_tree->Get_pointsCount()));ASSERT_( (_b >= 0) && (_b < ptrRef_tree->Get_pointsCount()));if( ptrNew_depth[ _a] != ptrNew_depth[ _b]){if( ptrNew_depth[ _a] < ptrNew_depth[ _b]){swap( _a, _b);}for( int i = power_range - 1; i >= 0; --i){if( ptrNew_father[ _a][ i] == -1){continue;}if( ptrNew_depth[ ptrNew_father[ _a][ i]] >= ptrNew_depth[ _b]){update( d1, d2, dist1[ _a][ i]);update( d1, d2, dist2[ _a][ i]);_a = ptrNew_father[ _a][ i];}}}ASSERT_( ptrNew_depth[ _a] == ptrNew_depth[ _b]);if( _a == _b){return {d1, d2};}for( int i = power_range - 1; i >= 0; --i){if( ptrNew_father[ _a][ i] != ptrNew_father[ _b][ i]){update( d1, d2, dist1[ _a][ i]);  update( d1, d2, dist2[ _a][ i]);update( d1, d2, dist1[ _b][ i]);  update( d1, d2, dist2[ _b][ i]);//--_a = ptrNew_father[ _a][ i];_b = ptrNew_father[ _b][ i];}}ASSERT_( ptrNew_father[ _a][ 0] == ptrNew_father[ _b][ 0]);update( d1, d2, dist1[ _a][ 0]);  update( d1, d2, dist2[ _a][ 0]);update( d1, d2, dist1[ _b][ 0]);  update( d1, d2, dist2[ _b][ 0]);return {d1, d2};
}
void update( int & d1, int & d2, int _w){if( _w > d1){d2 = d1;d1 = _w;}else if( _w != d1){d2 = max( _w, d2);}
}

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`Computer-Algorithm` 最小生成树MST,Prim,Kruskal,次小生成树

Contents

最小生成树

Algorithm

Prim

Code

Kruskal

Prim&KruskalPrim \& KruskalPrim&Kruskal算法的性质

最小生成树的性质

最小生成树的线性递增序列

例题

次小生成树

Code

例题

1148. 秘密的牛奶运输

356. 次小生成树

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