o 3ήc!6@sdZddlmZddlZddlmZgdZddZdd Z d d Z d d Z ddZ ddZ ddZddZddZddZddZddZededd d!Zedd"d#ZdS)$z Graph products. )productN)not_implemented_for)tensor_productcartesian_productlexicographic_productstrong_productpowerrooted_productcs fddttBDS)Ncs"i|] }|||fqS)get).0kd1d2r L/tmp/pip-target-vg8gfxp4/lib/python/networkx/algorithms/operators/product.py s"z!_dict_product..)setrr rr _dict_products rccs:t||D]\}}||ft|j||j|fVqdSN)rrnodes)GHuvr r r _node_products"rc cs|s.|s.|jddD]\}}}|jddD]\}}}||f||ft||fVqq|s^|r^|jddD]!\}}}|jdddD]\}}}}||f||f|t||fVqHq<|r|s|jdddD] \}}}}|jddD]\}}}||f||f|t||fVqyqm|r|r|jdddD](\}}} }|jdddD]\}}}}||f||f| |ft||fVqqdSdSdSNTdatarkeys is_multigraphedgesr rrrrcxydr jr r r_directed_edges_cross_edges."r*c cs|s.|s.|jddD]\}}}|jddD]\}}}||f||ft||fVqq|s^|r^|jddD]!\}}}|jdddD]\}}}}||f||f|t||fVqHq<|r|s|jdddD] \}}}}|jddD]\}}}||f||f|t||fVqyqm|r|r|jdddD](\}}} }|jdddD]\}}}}||f||f| |ft||fVqqdSdSdSrr!r$r r r_undirected_edges_cross_edges0r+r,ccs|r%|jdddD]\}}}}|D] }||f||f||fVqq dS|jddD]$\}}}|D]}|rD||f||fd|fVq2||f||f|fVq2q+dSNTrrr"r#)rrrrr r(r&r r r_edges_cross_nodesCsr/ccs|r%|D]}|jdddD]\}}}}||f||f||fVqqdS|D](}|jddD]\}}}|rD||f||fd|fVq/||f||f|fVq/q'dSr-r.)rrr&rrr r(r r r_nodes_cross_edgesQsr0ccs|r*|jdddD]\}}}}|D]}|D] }||f||f||fVqqq dS|jddD])\}}}|D]!}|D]}|rM||f||fd|fVq;||f||f|fVq;q7q0dSr-r.)rrrrr r(r&r'r r r_edges_cross_nodes_and_nodes_s$r1cCsT||ksd}t||s|rt}nt}|r(|}|S)Nz0G and H must be both directed or both undirected) is_directednx NetworkXErrorr" MultiGraphGraph to_directed)rrmsgGHr r r_init_product_graphos  r:cCsFt||}|t|||t|||s!|t|||S)aReturns the tensor product of G and H. The tensor product $P$ of the graphs $G$ and $H$ has a node set that is the tensor product of the node sets, $V(P)=V(G) \times V(H)$. $P$ has an edge $((u,v), (x,y))$ if and only if $(u,x)$ is an edge in $G$ and $(v,y)$ is an edge in $H$. Tensor product is sometimes also referred to as the categorical product, direct product, cardinal product or conjunction. Parameters ---------- G, H: graphs Networkx graphs. Returns ------- P: NetworkX graph The tensor product of G and H. P will be a multi-graph if either G or H is a multi-graph, will be a directed if G and H are directed, and undirected if G and H are undirected. Raises ------ NetworkXError If G and H are not both directed or both undirected. Notes ----- Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None. Examples -------- >>> G = nx.Graph() >>> H = nx.Graph() >>> G.add_node(0, a1=True) >>> H.add_node("a", a2="Spam") >>> P = nx.tensor_product(G, H) >>> list(P) [(0, 'a')] Edge attributes and edge keys (for multigraphs) are also copied to the new product graph )r:add_nodes_fromradd_edges_fromr*r2r,rrr9r r rr|s /rcC>t||}|t|||t|||t|||S)aReturns the Cartesian product of G and H. The Cartesian product $P$ of the graphs $G$ and $H$ has a node set that is the Cartesian product of the node sets, $V(P)=V(G) \times V(H)$. $P$ has an edge $((u,v),(x,y))$ if and only if either $u$ is equal to $x$ and both $v$ and $y$ are adjacent in $H$ or if $v$ is equal to $y$ and both $u$ and $x$ are adjacent in $G$. Parameters ---------- G, H: graphs Networkx graphs. Returns ------- P: NetworkX graph The Cartesian product of G and H. P will be a multi-graph if either G or H is a multi-graph. Will be a directed if G and H are directed, and undirected if G and H are undirected. Raises ------ NetworkXError If G and H are not both directed or both undirected. Notes ----- Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None. Examples -------- >>> G = nx.Graph() >>> H = nx.Graph() >>> G.add_node(0, a1=True) >>> H.add_node("a", a2="Spam") >>> P = nx.cartesian_product(G, H) >>> list(P) [(0, 'a')] Edge attributes and edge keys (for multigraphs) are also copied to the new product graph )r:r;rr<r/r0r=r r rrs ,rcCr>)aReturns the lexicographic product of G and H. The lexicographical product $P$ of the graphs $G$ and $H$ has a node set that is the Cartesian product of the node sets, $V(P)=V(G) \times V(H)$. $P$ has an edge $((u,v), (x,y))$ if and only if $(u,v)$ is an edge in $G$ or $u==v$ and $(x,y)$ is an edge in $H$. Parameters ---------- G, H: graphs Networkx graphs. Returns ------- P: NetworkX graph The Cartesian product of G and H. P will be a multi-graph if either G or H is a multi-graph. Will be a directed if G and H are directed, and undirected if G and H are undirected. Raises ------ NetworkXError If G and H are not both directed or both undirected. Notes ----- Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None. Examples -------- >>> G = nx.Graph() >>> H = nx.Graph() >>> G.add_node(0, a1=True) >>> H.add_node("a", a2="Spam") >>> P = nx.lexicographic_product(G, H) >>> list(P) [(0, 'a')] Edge attributes and edge keys (for multigraphs) are also copied to the new product graph )r:r;rr<r1r0r=r r rrs +rcCsft||}|t|||t|||t|||t|||s1|t|||S)aReturns the strong product of G and H. The strong product $P$ of the graphs $G$ and $H$ has a node set that is the Cartesian product of the node sets, $V(P)=V(G) \times V(H)$. $P$ has an edge $((u,v), (x,y))$ if and only if $u==v$ and $(x,y)$ is an edge in $H$, or $x==y$ and $(u,v)$ is an edge in $G$, or $(u,v)$ is an edge in $G$ and $(x,y)$ is an edge in $H$. Parameters ---------- G, H: graphs Networkx graphs. Returns ------- P: NetworkX graph The Cartesian product of G and H. P will be a multi-graph if either G or H is a multi-graph. Will be a directed if G and H are directed, and undirected if G and H are undirected. Raises ------ NetworkXError If G and H are not both directed or both undirected. Notes ----- Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None. Examples -------- >>> G = nx.Graph() >>> H = nx.Graph() >>> G.add_node(0, a1=True) >>> H.add_node("a", a2="Spam") >>> P = nx.strong_product(G, H) >>> list(P) [(0, 'a')] Edge attributes and edge keys (for multigraphs) are also copied to the new product graph ) r:r;rr<r0r/r*r2r,r=r r rrs -rdirected multigraphcs|dkrtdt}|||D]@i}d}|}|rG|}i}|D]}|kr,q%||vr;|||<|||q%||krAn|d7}|s|fdd|Dq|S)a0Returns the specified power of a graph. The $k$th power of a simple graph $G$, denoted $G^k$, is a graph on the same set of nodes in which two distinct nodes $u$ and $v$ are adjacent in $G^k$ if and only if the shortest path distance between $u$ and $v$ in $G$ is at most $k$. Parameters ---------- G : graph A NetworkX simple graph object. k : positive integer The power to which to raise the graph `G`. Returns ------- NetworkX simple graph `G` to the power `k`. Raises ------ ValueError If the exponent `k` is not positive. NetworkXNotImplemented If `G` is not a simple graph. Examples -------- The number of edges will never decrease when taking successive powers: >>> G = nx.path_graph(4) >>> list(nx.power(G, 2).edges) [(0, 1), (0, 2), (1, 2), (1, 3), (2, 3)] >>> list(nx.power(G, 3).edges) [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)] The `k`th power of a cycle graph on *n* nodes is the complete graph on *n* nodes, if `k` is at least ``n // 2``: >>> G = nx.cycle_graph(5) >>> H = nx.complete_graph(5) >>> nx.is_isomorphic(nx.power(G, 2), H) True >>> G = nx.cycle_graph(8) >>> H = nx.complete_graph(8) >>> nx.is_isomorphic(nx.power(G, 4), H) True References ---------- .. [1] J. A. Bondy, U. S. R. Murty, *Graph Theory*. Springer, 2008. Notes ----- This definition of "power graph" comes from Exercise 3.1.6 of *Graph Theory* by Bondy and Murty [1]_. rzk must be a positive integerc3s|]}|fVqdSrr )r nbrnr r szpower..) ValueErrorr3r6r;updater<)rr rseenlevel nextlevel thislevelrr rCrrQs0@  rcsbvr tdt}|t||fdd|D|fdd|D|S)aReturn the rooted product of graphs G and H rooted at root in H. A new graph is constructed representing the rooted product of the inputted graphs, G and H, with a root in H. A rooted product duplicates H for each nodes in G with the root of H corresponding to the node in G. Nodes are renamed as the direct product of G and H. The result is a subgraph of the cartesian product. Parameters ---------- G,H : graph A NetworkX graph root : node A node in H Returns ------- R : The rooted product of G and H with a specified root in H Notes ----- The nodes of R are the Cartesian Product of the nodes of G and H. The nodes of G and H are not relabeled. zroot must be a vertex in Hc3s(|]}|df|dffVqdSrrANr )r e)rootr rrEs&z!rooted_product..c3s6|]}D]}||df||dffVqqdSrL)r#)r grM)rr rrEs4)r3r4r6r;rr<r#)rrrNRr )rrNrr s r )__doc__ itertoolsrnetworkxr3networkx.utilsr__all__rrr*r,r/r0r1r:rrrrrr r r r rs,    7347 W