The Metropolitan Police committed "bad errors" and "missed opportunities" to stop the killer of Rachel Nickell before her death, a report has said.
Serial rapist Robert Napper admitted killing Ms Nickell on Wimbledon Common, south-west London in 1992.
The Independent Police Complaints Commission (IPCC) found Napper, 42, was identified as a threat to women in the mid-1980s.
The Met said it "regrets" that Napper "was able to carry out dreadful acts".
Colin Stagg, the man wrongly accused of murdering Ms Nickell said he was considering through his solicitor whether to pursue his own complaint through the IPCC.
His solicitor Alex Tribeck said his client wanted to ensure that the "professional negligence" of the Metropolitan Police should never be repeated.
But it said it did not intend to read out a statement of apology.
The IPCC said two other killings by Napper could have been prevented.
Samantha Bissett and her four-year-old daughter Jazmine were killed at their home in Plumstead, south London, in 1993.
Ms Nickell, 23, was sexually assaulted and stabbed 49 times by Napper in front of her two-year-old son Alex.
The IPCC says the Met apologised in private to Ms Nickell's partner, Andre Hanscombe, and to Alex but recommended it should now issue an "unreserved" public apology.
In a statement, the Met said Assistant Commissioner Cressida Dick will write to Alex and Mr Hanscombe to apologise again and she has offered him a face-to-face meeting.
"The Met has accepted that more could, and should have been done, and had more been done we could have been in a better position to have prevented very serious attacks by Napper," the statement added.
A Met spokeswoman later made clear that it regards the statement it issued this morning about the Rachel Nickell case as a "public apology", as called for by the IPCC.
But she said the Met does not intend to read out a statement of apology.
Reacting to the report, Mr Hanscombe said the IPCC shared his "sense of shock and disbelief" at the police errors.
He added: "Nothing is going to bring Rachel, Samantha or Jazmine back.
"But having had some time to come to terms with this new reality, I now believe the best way to serve those who paid most heavily is to make sure all the lessons have been learned, to make sure that this could never happen again."
Samantha Bisset's step-father, Jack Morrison, said he was not prepared to criticise the Met.
"We have a very good police force, but we can all make mistakes," he said.
"The police have been extremely kind to me and my family since Samantha's death, right up until the present time."
Rachel Cerfontyne, the IPCC Commissioner, said officers "inconceivably" eliminated Napper over a series of rapes on parkland in south London - known as the Green Chain rapes - because he was thought to be too tall.
She said: "It is clear that throughout the investigations into the Green Chain rapes and Rachel Nickell's death there was a catalogue of bad decisions and errors made by the Metropolitan Police.
"The police failed to sufficiently investigate after Napper's mother called police to report that he had confessed to her that he had raped a woman and, inconceivably, they eliminated Napper from inquiries into the Green Chain rapes because he was over 6ft tall.
"Without these errors, Robert Napper could have been off the streets before he killed Rachel Nickell and the Bissets, and before numerous women suffered violent sexual attacks at his hands."
Ms Cerfontyne said the mistakes by the force were "dreadful".
But she said no police officer will face disciplinary action because they had all retired, and one key senior detective has died.
Many of the mistakes had been publicised after Napper had been convicted.
But the IPCC said it had received fresh information from an ex-police officer that Napper had come to the attention of a police sergeant as a "serious threat to women" in the mid-1980s.
Another person said he contacted police two months after Rachel Nickell's death to say that he had overhead a conversation in a pub between Napper and a friend in which they were laughing about the killing.
A record on a police intelligence system in 2002 said Napper confessed to killing Rachel Nickell while he was detained in Broadmoor in 1997 or 1998.
He pleaded guilty to manslaughter on the grounds of diminished responsibility in December 2008.
Colin Stagg was cleared of her murder by the Old Bailey in 1994, after spending 13 months on remand.
He was subsequently awarded £706,000 in compensation from the Home Office in 2008 for wrongful arrest.